Elect infants

i) The eternal fate of those who die before they can exercise saving faith isn't an issue unique to Calvinism. For instance, John Wesley, in his Treatise on Baptism, says:

As to the grounds of it: If infants are guilty of original sin, then they are proper subjects of baptism; seeing, in the ordinary way, they cannot be saved, unless this be washed away by baptism. It has been already proved, that this original stain cleaves to every child of man; and that hereby they are children of wrath, and liable to eternal damnation. It is true, the Second Adam has found a remedy for the disease which came upon all by the offense of the first. But the benefit of this is to be received through the means which he hath appointed; through baptism in particular, which is the ordinary means he hath appointed for that purpose; and to which God hath tied us, though he may not have tied himself. Indeed, where it cannot be had, the case is different, but extraordinary cases do not make void a standing rule. This therefore is our First ground. Infants need to be washed from original sin; therefore they are proper subjects of baptism.

ii) For his part, Warfield summarizes no fewer than five different positions in Reformed historical theology, of which I'll comment on two:

Many held that faith and the promise are sure signs of election, and accordingly all believers and their children are certainly saved ; but that the luck of faith and the promise is an equally sure sign of reprobation, so that all the children of unbelievers, dying such, are equally certainly lost. 

More held that faith and the promise are certain signs of election, so that the salvation of believers' children is certain, while the lack of the promise only leaves us in ignorance of God's purpose; nevertheless that there is good ground for asserting that both election and reprobation have place in this unknown sphere. Accordingly they held that all the infants of believers, dying such, are saved, but that some of the infants of unbelievers, dying such, are lost. Warfield, Studies in Theology, 9:432-33.

i) It's not clear from this why some Reformed theologians tie the fate of dying infants to their parentage. This may be related to the argument for infant baptism, where parents sponsor their children. Or the notion that children of believing parents are in the covenant by virtue of their parentage–and thereby suitable baptismal candidates. In both cases we have a representative principle at work. 

There are, however, problems with tying the fate of dying infants to their parentage:

ii) What if one parent is elect, but the other is reprobate? How to split the difference? 

iii) Election can, and sometimes does, cut across family lies. The following combinations are possible, and actually play out in various cases:

a) Elect children of elect parents

b) Elect children of reprobate parents

c) Reprobate children of elect parents

d) Reprobate children of reprobate parents

Given that fact, it's unclear why some would argue that the eternal fate of dying infants is tied to the spiritual status of their parents. 

iv) It might be argued that God is more likely to save the children of believers. 

Elect infants, dying in infancy, are regenerated, and saved by Christ, through the Spirit, who worketh when, and where, and how he pleaseth: so also are all other elect persons who are incapable of being outwardly called by the ministry of the Word (WCF 10.3). 

i) I don't know for a fact why the Westminster Divines settled on this ambiguous formulation. Perhaps that's covered in the minutes of the Westminster Assembly. The historical question doesn't interest me that much. I'm guessing there were two reasons for the studied ambiguity. 

a) Since Scripture doesn't specifically address this issue, the Westminster Divines thought it best to be circumspect in how much they said on the subject.

b) As a consensus document, the Westminster Confession must sometimes finesse disagreements among different parties or individuals at the Westminster Assembly. 

ii) The formulation is committed to the existence of heavenbound dying infants. And the formulation is amendable to two additional, but opposing views:

a) All dying infants are elect

b) Some dying infants are reprobate

Both (a) and (b) are logically consistent with the Confessional wording. Neither (a) or (b) is logically entailed by the Confessional wording. Beyond a certain point, the Confession is noncommittal. It doesn't imply the salvation of all dying infants or the damnation of some dying infants. Rather, it leaves that an open question.

Many professing Christians, as well as many opponents of the Christian faith, find the whole subject of infant damnation morally appalling. This is sometimes caricatured as babies roasting in hell. I'll just make a few brief points:

i) It seems a bit ad hoc to claim that if Attila the Hun died at 5, he'd go to heaven–but if he died at 25, he'd go to hell. That makes damnation a misfortune of timing. 

ii) As I've discussed on several occasions, there's no reason to think hell is the same for all the damned. Dante popularized the notion of hell as physical torture, but that's a literary tradition. 

iii) Assuming (ex hypothesi) that some who die before the age of discretion are damned, that doesn't mean they remain in the psychological condition in which they died. There's no reason go think death freezes the decedent in the physical or mental condition he was in at the time of death. To take a comparison, if a Christian dies in a state of advanced senile dementia, that hardly means he will be senile for all eternity. Heaven is restorative. 

By the same token, if Attila the Hun died at 5 and went to hell, I take that to mean that he'd mature psychologically. But he'd mature without common grace or special grace. His eternal condition would be characterized by the absence of grace. There'd be nothing to mitigate his sinful predisposition. It doesn't require any external punitive environment. Rather, it's a deprivation.