Go with an open Glock.
I built one for my 15 y/o daughter (5'3" and 115lbs) and she loves it! You can get a new or slightly used G17 for about $425, add a SJC convertible comp, a slide racker, mag well, 11lb spring, extended mag release and 3 mags with Arredondo extensions (23rds)for roughly $1200. It shots minor well, both handloads and any factory rounds you feed it. No jams, etc. Damn thing just runs, it is a glock after all. She likes the dot and can manage to safely navigate courses of fire, which is the most important thing IMHO. If her skills keep growing, I will look at moving her up to a full house open gun in the future, once done with the Glock, I am sure I can get most of my money back by parting it out.
Let me know if you need more info, I would be glad to help out.
Roger