This is my youngest son. He is small for his age, currently about 5'. The salesman recommended what he called an Adult Small (he didn''t want to specify e.g. 24" because he said each bike is built differently), which he said would last him until about 5' 7" by adjusting the seat as
@MoYS mentioned, which should be a good 3-5 years.
The salesman did also mention that he does much more maintenance on kids' bikes than adult bikes, because kids tent not to take such good care of their bikes, which is another reason I was inclined to try to go cheaper. Any recommendations of decently-made bikes for a reasonable price from a big-box store?
The salesman said they charge $80 just to assemble a bike, which eats into a lot of the savings, though don't some stores include assembly when you buy from them? Unfortunately I'm not the least bit handy, and my son is better than me, but I don't know that it's realistic for him to do high-level bike maintenance.