"Dick" is a mascot of sorts on their signage, I think it may be a caricature of their founder, but I got nothin' to back that up.
Meanwhile, it's incredibly difficult to say you want to eat there without sounding inappropriate. "Man, I could go for some Dick's tonight" and "So who wants Dick's for lunch?" being prime examples. But I guess that's the point. (For the record: the food's actually good, at least last time I went there.)