PWM PMIC for hacking up a small display?

[joeyjoejoe]

I got some SWAG that is a simple LED-backlit case that let's you put in letter and customize a message. Inside is simply two LED trips, a 10Ohm resistor and it runs of 3xAAA.

I'd like to hack it up to get a bit more life, and possibly make it run off a single 18650 with a simple protection circuit and microUSB charging. Also, ditch the resistor for something more power efficient.

I could do a simple 555, but it got me to thinking.. this seems like a straightforward application. Are there existing IC's that have a PWM output, can drive LED's and have a low current? Do they have a name I can search for? Category in Digikey?

(I tried some google-fu for that but got mostly complete LED driver circuit boards or complete LED drivers themselves...)