Age? Price range? That would help narrow it down, but it's hard to beat a Marlin Mod.60, or a Ruger 10/22. But depending on age/size those may be too large for starting out, might need a lil' 22 crickett.
Figured I would throw this out, not sure how relevant given the age difference. I own a buckmark .22 and am happy with it. I brought my nephew out shooting when he was 5 and while he did enjoy shooting the buckmark, it is a bit too heavy for his little hands. So I ended up buying a Phoenix Arms .22 for him to use. Small, light, and not very expensive. For around $150 I have the pistol with standard barrel, and a 5" barrel. It works well. The only failures with the gun I can attribute to him limp wristing it. I have only used cheap bargain pack shells through it, and have avoided anything with a little extra kick because it is a cheap pot metal gun and is supposedly not made for high pressure rounds.
Again, just sharing my experience, do as you will.
"ultra...good" I have been looking at the longer barrel phoenix but was little afraid of the Quality.
Does anyone know anything about the Beretta Neos?
My 14 year old started out with my old Smith Mod. 15 shooting powder puff loads. She shoots my Sig in .40 now and handles it well. I bought a Ruger MK III 22/45 for my 12 year old because her hands are a little small for the Hogue grips on my Smith. I bought it very gently used for $230. She shoots it well and I couldn't be happier. Quite frankly I have a blast with it as well. My buddy has a Walther that she she shoots just as well with. Both would fit in your price range but there are plenty more out there, these are what she has used. Take it for what it's worth.
My wife has a Buckmark, and loves it. My son loves shooting my GSG 1911, both of those I purchased (used) under the $300 limit you are looking for. My personal favorite .22 handgun that I own is my heritage revolver. It was well under the limit your looking to stay under new, has been very reliable, and pretty accurate even with the older style fixed sights that mine has. The other plus to the 22 revolvers is that the grips on many are smaller, and might be easier for a younger lady to hold onto.
I have a friend who has a heritage rough rider, it was fun to shoot but slow to reload. I would rather her learn on a semi auto platform.
Thanks for the advice!