I thank God, that I have never been homeless. So I'll share some thoughts for consideration, or discussion, all of which might be invalid.
My son and his friend did some hiking on the Appalachian trail, after the first day on the trail, his friend left a lot of gear at the shelter they stayed the first night. Too much stuff... too much weight.
I read a book of one man's journey on the trail. "no rain... no Maine" is a standard saying along the trail. He also spoke of a vietnam vet, who after a couple of days at the beginning of the trail stopped at an outfitter - who dumped his pack and set him up with a lot less stuff, dual purpose stuff, and lighter weight.
Another story, while on the trail, he and another were offered a can of Spam. He turned it down (his partner for that stretch gladly accepted it) at the end of the day, his partner opened the can, but of course couldn't eat it all in one sitting. He gave the rest to the author. His strategy all the time was to let his friend carry it. Again... the point is that even ounces on your back at the end of the day, add up.
All this to say... if you are carrying your life on your back, you want it to be light.
Considering the other more essentials:
food
water
shelter
fire
clothing (considered in part, shelter).
food procurement (animal trapping)?
Picking a light would depend to a great extent (like someone else mentioned) would depend upon the environment, urban, suburban, rural, wooded.
I might consider following reppans suggestions for a single AA cell Quark XML.
On the other end of the consideration, I might get one of those small 1 inch x 2 1/2 inch - one 5mm led solar charged lights.
If I were gifting the light to someone, I might give him the Quark, OTHO if it were me, I might get the little solar light, and put the rest of the money on a VISA card for food later on. I guess it all depends.
Regarding charging options, someone once mentioned that in a suburban environment many people have solar powered lawn lights lighting the path to their yards etc. Often they have low ma NiMH AA batteries in them. I suppose that you could sneak by early in the evening, after they have been charging all day, and didn't have a chance to run too long, and swap in dead batteries. That would be a little more honorable than out-right stealing.
Homeless, with a smart-phone, and a power bank? Unless you have a PO box, or other secure place to store your gadgets, IDK...
If you can have library access, they'll always have power outlets, and internet access to card holders, (I don't know if they are universal, ie, if you have a card in Iowa, can you use it in Florida).