I'd go with a Weihrauch HW25 Carbine in .177 with a simple 4x or 6x fixed air gun scope.
It's a nice quality but inexpensive rifle of the right size / weight with a good trigger, plus it's accurate - nothing is more off-putting to a beginner, particularly a child, than a gun that's horrible to use or is inaccurate.
If you want to push the boat out and buy a pre-charged air rifle, I'd suggest a secondhand Air Arms S200 Mk1, which is what many air gun clubs use for beginners and younger members.
Having been here with both of my sons, I'd also recommend a few reactive targets that can be knocked down and re-set by a pull-string, some spinner targets and a few homemade ones such as water-filled balloons, eggs etc., as these definitely hold a youngster's interest longer than punching holes in paper.
A.