According to one source, the .22 LR was developed by Stevens in 1887, and came out in 1888. The rifle that used it would have been the Stevens Tip-Up, which was introduced in the 1870's. (Note that the .22 LR could be fired in rifles chambered for the .22 Long, but most .22 Long revolver cylinders were too short, hence ".22 Long RIFLE".
The Winchester Single shot came out in 1885, but first chamberings were for center fire cartridges, and I don't know when it came out in .22. It would have been expensive, and only for a VERY good boy.
The Remington Model 2 Sporting Rifle (a rolling block action) came out around 1873 and definitely was made in .22 Long by 1888. The better known No. 4 did not come out until 1890, but was definitely made in .22 LR specifically. These were good quality and sold at a reasonable price.
The Marlin-Ballard No. 3 Gallery Rifle was made from c. 1876 to 1891 in several variations, including fancy wood and a pronged buttplate. These were also expensive.
Henry Marcus Quackenbush began making .22 rifles in 1871; they were of lesser quality, but inexpensive and sold well. They were often used as prizes for selling salve or other door-to-door products often peddled by boys. (Another product of the company was nut crackers and HMQ is still seen on the common spring type, unless they are all made in China now.)
I have concentrated on single shot rifles because I think that would be more likely for a boy than a repeater; further, repeaters in .22 were pretty uncommon and expensive. Other calibers would have been possible. Several .25 rimfire cartridges were available around that time, as well .32 and .38 rimfire and centerfire. But I agree that .22 would be more likely for a boy.
Jim
Jim
