It would probably be good to have an external 12-19V input to a separate charger chip, which will enable it to be charged from a laptop PSU easily, and will satisfy most users. Charge time on that would be under 2 hours, and it would be usable while charging ( though at the expense of higher output noise though) without killing USB power delivery, allowing use on a unpowered hub ( common laptop limited ports).
2A USB power supplies are very common nowadays given that a lot of tablets use them. Also, 900mA USB 3.0 ports are becoming more common, along with high current (1A or more) USB ports on some new motherboards.
Well "hybrid" NiMH is what the inventors Sanyo marketed them as ,others companies are marketing them also now, some as low self discharge or "LSD NIMH" , (which are mostly sanyo's batteries rebranded, according to some websites).
That's a NiMH chemistry tweaked for hybrid vehicles and other high peak current, low self discharge applications. It is more expensive than regular NiMH for the same capacity.
LiFePO4 would be nice, but the charger chips are not cheap and plentiful like LiIon/LiPoly
LiFePO4 is much more forgiving than common Li-ion (overcharge would just greatly shorten the life instead of causing thermal runaway), so you can implement the charging and discharge cut off in software. Back it up with the watchdog timer and it should work quite well.