Quite a few options out there that in general would be a better choice than a lever action 30-30. If you want your daughter to really like hunting and shooting, start with a better option, she can always get one later if she like that style.
7mm-08 has been listed by several informed posters, I'd listen to them. Pick up a Savage Axis II with the Accutrigger and get some Hornaday Custom Lite cartridges for it. She will be all set for a lifetime of hunting as she can always move up to full power loads for bigger game if she wants, or use these even for 'yotes. It's what I started my 10 year old son with. He's now 13 and truly in love with that rifle. That's what you want for your daughter.
If you want another fantastic deer-sized game caliber, look at a 25.06. More power than a .243 with the same light kick, but with these two calibers, you can't go up in power like with the 7mm-08. With the reduced recoil loads, the 7mm-08 is almost identical in recoil as the previous calibers.