At that height and size, I would recommend something like a Haro F1c or a Mongoose Mini Motivator as these have smaller frames than the rest of the standard 20"BMX's
It really depends on how deep your pockets are, because otherwise you could get a decent 18" wheel bmx
BMXes are pretty much one size fits all if you're buying a 20" wheel model and it will be too big. 20" means 170mm cranks. I suggest either the smaller wheeled BMX that Neil suggests (check the cranks are 130-140mm or thereabouts) or a 20" MTB. It won't be long before he's onto a 24" MTB.