Repurpose an old lamp with LEDs. Where to start
I'd break this down in the opposite direction. This isn't a full solution, but the two parts can be addressed with googling or looking elsewhere on this site.

1) Starting from the USB charger: that delivers 5v.

2) Battery: Lithium polymer batteries have a 3.6V-4.2V range, and ICs and circuits designed to charge them from USB 5V are widely available. This sort of thing (not necessarily the cheapest option). Make sure your design checks the battery's thermistor.

NiCd and NiMH are the other common battery options, but they also require special circuitry. Lead acid is easiest to charge but not very portable.

3) LED and driver: you could have one big one or lots of small ones. If you have lots of small ones you might be able to drive them directly from the battery (their forward voltage is about the same as the battery voltage), but this is non-optimal. Have a read of this paper from Maxim. Again, there's already lots of stuff on electronics.stackexchange.com about "how to drive LEDs".

If you go for one big one it will definitely need a charge pump based controller of suitable current rating. If it's really big then you need to think about cooling it.