G43X is a strong default, but have her hold a **Sig P365** and a **S&W M&P Shield Plus** before committing. All three are in-budget, all three are reasonable first pistols, and for small hands the grip contouring varies enough that one will feel obviously better than the others to her specifically.
The other "budget first pistol" to at least consider: **Ruger Max-9**. Legitimately capable sub-$450 micro-9, puts her 30-40% under budget for ammo/lessons. Not as refined as the 43X/P365 but it's a very serviceable carry pistol and the price difference is significant if she'll shoot a lot.
My actual recommendation for a new carrier: whichever of the three she handled and *wanted* to shoot. Fit and comfort drive practice volume, practice volume drives skill, and skill is the whole point.