To answer your question, until today I would have said "Captions", with the green icon, more features than the one with the blue icon, by BigBluePixel.
However as of today I'd have to recommend "Skitch" by the folks who produce Evernote. Skitch is free, does not have as many features as Captions, but will do what you were asking for, that is let you write your name on a photo, plus a lot more.
I'm sure I'll be using Skitch a lot for quick captions on photos or screen captures, but when I want more and better options I'll use the green icon version of Capture.
So to answer your question, I don't think there is one single "best" app, but Skitch for free, and Capture for $$s, would be my choices.