Button quail are excellent pets, but not the fuzzy hold me type... they are pretty and not as difficult to raise as exotic-er varieties.
Coturnix quail are the BEST egg layers, and grow real fast. It only takes 8 weeks from hatch to start laying!
Very easy to raise and hatch.
Bobwhite quail are regulated by some states as they are a native bird... so you'd have to pay for a gamebird license. They are also more finicky and beat each other up on a regular basis.
Gambel/Mountain/Texas Bluescale/Valley quail are all difficult to raise/hatch, brood, and take forever to mature just like the bobwhite.
If I were to recommend a type, I would say Coturnix. Just sayin'!!! Good luck!