To some extent this is a problem for everyone. Catholics agree with Calvinists that those who are elect are the ones who are eschatologically saved: who will go to heaven in the afterlife (cf. Catechism of the Catholic Church: #1045: "The beatific vision, in which God opens himself in an inexhaustible way to the elect, will be the ever-flowing well-spring of happiness, peace, and mutual communion."). We, too, believe in the election of the righteous, or those saved in the end (both Thomists and Molinists accept this), or predestination (though we hold it in a paradoxical "both/and" tension with free will). So in that sense, the Catholic, too, has to explain these references that include Judas, as somehow an exception to the rule. It's "weird" any way we look at it.
The relevant difference between the two systems in this regard is that Calvinists (unlike Catholics) believe in double predestination, or the predestination of the damned / reprobate as well as the saved. So the particular Calvinist difficulty is to explain how Judas could be described in a class of those who are "elect" if in fact he was predestined from eternity to be damned. It seems that (granting these presuppositions) he would never have been described that way at all.
But the Catholic can more easily say that there might perhaps be some conditionality to the term "elect" -- at least in some cases, since we (along with Protestant Arminians, Wesleyans, etc.) believe that a person can lose justification and salvation, should they decide to consciously turn away from and reject the God they once served. Perhaps then (I'm not asserting this but simply thinking aloud), "elect" in Judas' case is analogous to a person who is justified and then loses his justification and right standing with God (and ultimately, salvation) due to sin and rebellion. Before we examine this matter further, let's take a look at the relevant biblical passages (RSV):
(Strong's word #1588)
Matthew 22:14 For many are called, but few are chosen.
Matthew 24:22,24,31 And if those days had not been shortened, no human being would be saved; but for the sake of the elect those days will be shortened. . . .  For false Christs and false prophets will arise and show great signs and wonders, so as to lead astray, if possible, even the elect. . . .  and he will send out his angels with a loud trumpet call, and they will gather his elect from the four winds, from one end of heaven to the other. (cf. Mark 13:20,22,27)
Luke 18:7 And will not God vindicate his elect, who cry to him day and night? Will he delay long over them?
Romans 8:33 Who shall bring any charge against God's elect? It is God who justifies;
Romans 16:13 Greet Rufus, eminent in the Lord, also his mother and mine.
Colossians 3:12 Put on then, as God's chosen ones, holy and beloved, compassion, kindness, lowliness, meekness, and patience,
2 Timothy 2:10 Therefore I endure everything for the sake of the elect, that they also may obtain salvation in Christ Jesus with its eternal glory.
Titus 1:1 Paul, a servant of God and an apostle of Jesus Christ, to further the faith of God's elect and their knowledge of the truth which accords with godliness,
1 Peter 1:1-2 Peter, an apostle of Jesus Christ, . . .  chosen and destined by God the Father and sanctified by the Spirit . . .
1 Peter 2:9 But you are a chosen race, a royal priesthood, a holy nation, God's own people, that you may declare the wonderful deeds of him who called you out of darkness into his marvelous light.
2 John 1:1 The elder to the elect lady and her children, whom I love in the truth, and not only I but also all who know the truth,
2 John 1:13 The children of your elect sister greet you.
Revelation 17:14 they will make war on the Lamb, and the Lamb will conquer them, for he is Lord of lords and King of kings, and those with him are called and chosen and faithful.
(Strong's word #1589)
(Strong's word #1589)
Acts 9:15 But the Lord said to him, "Go, for he is a chosen instrument of mine to carry my name before the Gentiles and kings and the sons of Israel;
Romans 9:11 though they were not yet born and had done nothing either good or bad, in order that God's purpose of election might continue, not because of works but because of his call,
Romans 11:5 So too at the present time there is a remnant, chosen by grace. . . .  What then? Israel failed to obtain what it sought. The elect obtained it, but the rest were hardened,
Romans 11:28-29 As regards the gospel they are enemies of God, for your sake; but as regards election they are beloved for the sake of their forefathers.  For the gifts and the call of God are irrevocable.
1 Thessalonians 1:4 For we know, brethren beloved by God, that he has chosen you;
2 Peter 1:10 Therefore, brethren, be the more zealous to confirm your call and election, for if you do this you will never fall;
(Strong's word #4899)
(Strong's word #4899)
1 Peter 5:13 She who is at Babylon, who is likewise chosen, sends you greetings; and so does my son Mark.
(Strong's word #1586)
(Strong's word #1586)
Mark 13:20 And if the Lord had not shortened the days, no human being would be saved; but for the sake of the elect, whom he chose, he shortened the days.It's fairly obvious that all these words are synonymous. Scripture usually brings out its own intended meanings via cross-referencing of this sort. Eklektos and eklegomai both appear in a single passage:
Luke 6:13 And when it was day, he called his disciples, and chose from them twelve, whom he named apostles;
John 6:70-71 Jesus answered them, "Did I not choose you, the twelve, and one of you is a devil?"  He spoke of Judas the son of Simon Iscariot, for he, one of the twelve, was to betray him. (cf. 6:64: ". . . Jesus knew from the first who those were that did not believe, and who it was that would betray him.")
John 13:18 I am not speaking of you all; I know whom I have chosen; it is that the scripture may be fulfilled, `He who ate my bread has lifted his heel against me.'
John 15:16,19 You did not choose me, but I chose you and appointed you . . .  . . . I chose you out of the world, therefore the world hates you.
Acts 1:2 . . . the apostles whom he had chosen.
Acts 1:24 And they prayed and said, "Lord, who knowest the hearts of all men, show which one of these two thou hast chosen
Ephesians 1:4 even as he chose us in him before the foundation of the world, that we should be holy and blameless before him.
Mark 13:20 And if the Lord had not shortened the days, no human being would be saved; but for the sake of the elect [eklektos], whom he chose [eklegomai], he shortened the days.The "elect" are, simply, the ones whom Jesus has "chosen." Other Scriptures show that this election is from eternity (1 Peter 1:2; Eph 1:4; cf. Rom 8:29-30: "foreknew"; "predestined"; "called"). The immediate difficulty stems from those passages under eklegomai above that include Judas as one of the twelve elected / chosen disciples (Lk 6:13; Jn 6:70 -- it mentions Judas in particular --; Jn 15:16,19). John 13:18 implies (?) that Judas was not chosen, using the same Greek word, which means that there are either multiple applications of the word (as is usually the case in Scripture) or that Jesus contradicted Himself (which most observant Christians would reject), or that there was a later interpolation not attributable to Jesus Himself (rejected also by most biblical exegetes who accept, as I do, the text of the Bible at face value). John Calvin recognizes that it is at least potentially a problem for his own system of double predestination and provides his own answer for this:
In elsewhere numbering Judas among the elect, though he was a devil (John 6:70), he refers only to the apostolical office, which though a bright manifestation of divine favor (as Paul so often acknowledges it to be in his own person), does not, however, contain within itself the hope of eternal salvation. Judas, therefore, when he discharged the office of Apostle perfidiously, might have been worse than a devil; but not one of those whom Christ has once ingrafted into his body will he ever permit to perish, for in securing their salvation, he will perform what he has promised; that is, exert a divine power greater than all (John 10:28). For when he says, “Those that thou gavest me I have kept, and none of them is lost but the son of perdition,” (John 17:12), the expression, though there is a catachresis in it, is not at all ambiguous. The sum is, that God by gratuitous adoption forms those whom he wishes to have for sons; but that the intrinsic cause is in himself, because he is contented with his secret pleasure.This is interesting, since Calvin states that even an apostle may eventually be lost. That means that (in his system) he was predestined to be lost (it can't be otherwise for Calvin and Calvinists), in which case he was never elected at all (yet Scripture says he was), and therefore could not possibly do any good thing of his own volition (according to total depravity). I respond to this: "if an apostle -- chosen by God Himself -- is not necessarily elect, then how can we know that anyone who is (ostensibly) justified or regenerated will persevere to the end?" And that has implications for the entire Calvinist theological edifice of TULIP, imputed justification, etc. (though in fairness, I note that Calvin taught that we cannot know for sure who is and is not of the elect).
(Institutes of the Christian Religion, Book III, Chapter 22, Chapter 7)
The fact remains that one whom God "elected" fell away. Scripture doesn't say that Judas was predestined for reprobation from eternity. It simply says that Jesus "chose" (eklegomai) him. He was chosen in the same sense as the other disciples, so this couldn't have been in the sense of predestined reprobation (since the others appear to have been of the elect in the standard sense of "saved"). As I said above, these difficulties are not just Calvinist ones. Catholics also have to consistently incorporate this data. But I think Calvinists have relatively more difficulty because of how they construe double predestination and reject the notion that anyone could be truly chosen and in the fold, and then fall away (perseverance of the saints). Calvin raises another interesting question as well, by mentioning John 17:12:
John 17:11-12 And now I am no more in the world, but they are in the world, and I am coming to thee. Holy Father, keep them in thy name, which thou hast given me, that they may be one, even as we are one.  While I was with them, I kept them in thy name, which thou hast given me; I have guarded them, and none of them is lost but the son of perdition, that the scripture might be fulfilled.So now we have God the Father in on the election process (which would follow anyway, as His will and the Son's are unified at all times). If one of the very disciples can be "elected" and "given" by God the Father and called, yet fall away and be lost, then is this not troublesome for the Calvinist system, which holds that God predestines from eternity wholly apart from men's free will decisions, and that no man can overcome that; he cannot fall away, once having been so called and predestined? We don't have that difficulty because we accept both free will and the possibility of apostasy (while we reject Pelagianism and semi-Pelagianism: the notions that men can save themselves, as opposed to salvation being totally from God's grace: sola gratia).
Perhaps Calvin or Calvinists (following Jesus Himself in Jn 13:18 and 17:12) would point out that this was so that Scripture and prophecy would be fulfilled (Jesus was to be betrayed), but that (while true) doesn't alleviate the difficulty because we can then retort that if this is the case here, why could it not be in any number of situations, in God's providence, where He incorporates men's free will decisions of good or ill into His plan? What is to stop anyone from concluding that if this includes Judas' defection, "that the scripture might be fulfilled", why could not many other conceivable situations of apostasy, "that the scripture might be fulfilled"? It's true that exceptions don't usually disprove the rules they contradict, but too many exceptions can cast doubt at least upon the universality or sole application of a supposed rule in one fashion. Again, I am mostly speculating; not taking any ironclad position. Mostly I find this interesting to ponder.
Another fascinating line of inquiry has to do with the angels. St. Paul states:
1 Timothy 5:21 In the presence of God and of Christ Jesus and of the elect [eklektos] angels I charge you to keep these rules without favor, doing nothing from partiality.If God elected angels, then He must have not elected the fallen angels. But if even the fall of Satan and his demons was predestined by God, so that it couldn't have been otherwise (just as Calvinists say God does with the reprobate), then they had no free will either. And this has momentous implications. The Westminster Confession, adhered to by many Calvinists (especially Presbyterians) seems to assert this (to Catholic ears, rather extreme, bizarre, and troubling) state of affairs:
I. God from all eternity, did, by the most wise and holy counsel of His own will, freely, and unchangeably ordain whatsoever comes to pass; . . .If God predestined the very existence of evil, entailing that Satan didn't freely rebel, but did so because God predestined it, and it couldn't have been otherwise, then it seems that God would be the author of evil. Using the Calvinist understanding of election, all of this, I think, would follow. But under the different Catholic / Arminian / Wesleyan / post-Luther Lutheran systems of (non-Pelagian) free will, this might be construed as another "exceptional" usage of "elect" and not the norm. If it is an exception to the rule, as with the Judas example, then we now have two, leading one to suspect that additional such applications might be possible, and that the rule is much weakened, by virtue of two obvious and major exceptions.
III. By the decree of God, for the manifestation of His glory, some men and angels are predestinated unto everlasting life; and others foreordained to everlasting death.
IV. These angels and men, thus predestinated, and foreordained, are particularly and unchangeably designed, and their number so certain and definite, that it cannot be either increased or diminished.
(Chapter III: Of God's Eternal Decree)
A possible solution, I submit, is the position that I myself hold. In the Molinist understanding of predestination, God takes into account future free will actions of creatures, in His Middle Knowledge (scientia media) and this is a factor in His election or predestination. This doesn't account for the Judas anomaly, but it could account for the phrase elect angels" in a way that doesn't require God predestining the fall of Satan and his demons from eternity, rather than their free choice being the cause.
I welcome discussion on this, as always. As in all issues related to the deep mystery of predestination, I don't claim to have all the answers (not even close), but I think this is a fascinating biblical motif to ponder.