What kind of setup can take portable solar USB power & charge a AA battery charger that is usually designed to plug into a wall?
You're approaching this from the wrong end:

What you're trying to do is:

solar cell -> step down or down -> USB (5.0V) -> step up -> cigarette plug (12V) -> step up -> wall plug (120V or so) -> step down -> NiMH cell

That's ... very bad. As every step up and step down wastes energy, this will hardly work.

Instead, just use a charger that works directly off 5V (since I guess you can't access the solar cell's output directly).

Chips like Maxim's DS2710 have been around forever. There's plenty of even cheaper alternatives – but using the DS2710 would kind of feel like the "least effort, works instantly" alternative for designing this.

I'm also pretty sure you can buy charger devices that you can plug into USB directly. Every solution's bound to be cheaper than your convert-all-over-the-place approach.