I am sort of a fan of Air rifles. I taught all three daughters about gun safety starting with .177 Pellet, then .22LR, then 12 GA Rem 870. They all know how to shoot, clean and disassemble but none have chosen shooting as a hobby.
I own an RWS Model 34 "Diana" which is a nice gun.
Amazon.com: Diana RWS 34 Meisterschutze Pro Compact air rifle: Sports & Outdoors
It is probably heavy for a 11 yr old though. We used to live on a rural property that backed up to a 100 yr old Walnut grove teeming with rats. At last count I had well over 50 confirmed kills with the 34 and a night sight. Pretty fun shooting actually.
For a first air rifle I like these:
Amazon.com: Gamo Hornet Air Rifle, .177 Caliber: Sports & Outdoors
Break barrel, accurate, 4x scope, light weight at 6lbs. Build or buy a nice pellet trap to contain your shots and enjoy even at close range.