Hi! I'm new to this site. I found this thread interesting and figured I would comment.
I do believe that stats should be largely dependent on the specialization that they desire, and you would need to balance their stats on an individual basis. That being said, I do think that a Dex bonus, and a max strength per level should be enacted, with a judgement call on your part as to the actual limit. This should make sure they function at a slightly lower level, and require them to think and plan more. Being a child is a liability, in general.
A template that could work is this:
HD: 1d6-2 Min 1 Max 5 (allow Barbarian 1d8-3 max 7)
Ability Modifiers:
Strength: Normal max 10 (for fighter youngsters 14) with 3/4 weight capacity.
Dexterity: +2
Constitution: +2 (note max health per HD, mostly for saves and to indicate healthier children)
Intelligence: Dependent on class.
Wisdom: Max 14
Charisma: Dependent on class.
I would ignore racial bonuses entirely. Let them choose a race, sure, but don't count the bonuses unless they wish to trade in for the standard adult of that race and take an adult age. Considering the adaptable model given I find this to be suitable.
Allow bonuses to be traded around. For instance, the wizard might want the Con and Cha bonus on Int instead, considering the focus for the prodigy I would agree. Use Wizard HD in that case.
Another option, is to allow a very limited point buy, say, perhaps 4 points total to spread while counting the bonuses listed (with all stats marked 10 to start), with no more than 2 of the four to be allowed in a single stat, and enforce max limitation. I would count them as medium creatures, unless they intend to be exceptionally young or relatively short for a human. (Consider that hundreds of years ago the average height was not the same as now, if you wish to consider the D&D world as in the past) Although, I would grant them the bonus to hide, a bonus to climb, and a bonus to AC but not to hit.
I know it seems like there are a lot of bonuses I am tossing out, but considering the limitations I find them to be pretty fair.
For over 14 I would use normal stats, considering that many differences wouldn't factor in well. I have met some exceptionally strong 14 year olds. Of course there are extremely flexible people in that age group, and very bright minds aren't really dictated by age. Anyone can be charismatic, and given circumstances some younger people can be fairly wise. More so than many adults given that these are prodigies we are talking about.
I do stick by enforcing max stats for children under 12 and a bumped limit to the middle years. As a 10 year old kid there are certain aspects of your physicality that only puberty will be able to change. Granted, they *may* be extreme outliers, like a 6' tall 10 year old, but I wouldn't enforce that.
I would suggest using a point buy system (if you don't want to use the suggestion listed above), but make it a little more limited. For instance, if you normally did the "Elite" point buy system, bump it down one. This is to make the limitations on scores less of a burden, and more just an aspect of a character.
I would also give extra skill points, considering how much easier it is for a child to learn than an adult, perhaps 2 extra per level, (no x4 at first) and drop the +2 to +1 when they add the total amount the x4 would have added. This will help counter-act the limitations set on some of their stats, and depict some growing up on their part. I know it is a pain to keep track of, so just as everything else suggested it is entirely optional. In all honesty, I am not sure that I would encourage fighter type classes, but would focus on the dexterity, mentally, or charisma based classes instead.
I would also (for under 12) make them buy the small variants for armor and weapons, perhaps aside from light weapons, daggers, short swords, and the like.
I might re-read this latter and think "What the heck was I thinking?!?", but I would like to think I did a decent enough job.