If I was going to take my daughter hunting, I'm obviously limited to the contents of my gun safe as to what I could offer to her. But fortunately that selection would be easy: My Ruger No. 1A in 6.5x55 Swede (which I've already taken two deer with this season) is small in size but has a good solid heft to it. It's super simple to operate (a lever, a breech load & a tang safety). It has mild recoil and so far every deer I've shot with it has expired within visual sight of where I shot it so tracking has been easy. A youth could hunt with it and I'm still putting deer in the freezer with it at 62 so it's something she'll never outgrow.

But this advice is based on my limited experience & since I have a fondness for ambidextrous single-shot falling-blocks, it reflects my personal bias too. Mule Deer's advice to let your daughter try a variety of rifles if possible is great advice and it would probably be a more enjoyable way of acquiring the rifle too. While there have been a lot of good suggestions in this thread, for the hunting I do, I find the 6.5mm projectile to be a really good size selection (i.e. not too small / not too big) for the majority of the hunting I do. YMMV