It depends on what kind of riding he wants to do... a 24'' is generally OK for racing, but is considered a "Cruiser" as most of them are extremely difficult to do tricks on.

If he's already been riding for 3 years he probably has a good idea of his want's and needs. To me it sounds more like he's ready to move up to a better quality more professional BMX.

I'm 6'2 and I can ride a 20.5 TT very comfortably, but that's because I grew up on 18''TT's. 21'' maybe the way to go for him, but it really depends on the geometry of the frame, and yes the size of the handle bars.