Learn English – When to double the consonant before the suffix “-able”

Are they still teaching the old 'long/short' vowels? If so, here's the rule:

If the syllable before the /-ing/ is pronounced with a 'long' vowel, leave the final consonant single (and delete any final silent /e/)
If it's pronounced with a 'short' vowel, double the final consonant.

It may help make this clearer if you explain that a vowel before a doubled consonant is (almost) always short. Then write the /-ing/ forms out 'wrong' and invite your son to pronounce them the way they look:

As Renan points out, it gets more complicated when the final syllable of the base form is unstressed; but it looks like your son hasn't gotten that far yet.

As reflected in comments, there's no real "rule" here (though there's a tendency for -or to occur more often in words with Latin roots). So basically, you just have to learn them.

But things aren't as bad as they appear. Not only is the -er form more common in established words - it's far more "productive" for new terms. Also, as RegDwight points out in this ELU answer on the subject, there are many words where either spelling is acceptable (adviser/advisor, convener/convenor, etc.).

So instead of having to learn every word separately, all you have to do is remember those where only -or is acceptable (which as of today, includes administrator).

In short, slow as it might be, the general trend is towards -er. Adopt that as your default, and with any luck by the time you need one of today's more obscure "-or - only" words, the -er form will be acceptable!

EDIT: As @Anixx correctly points out, strictly speaking there is no currently productive suffix -ter in Modern English (the only instances where it's recognized as a meaningful "morphemic element" are laughter::laugh and slaughter::slay). This question and answer address the -or / -er distinction.