Most of the time, when your child is wishing for a horse, they’re not thinking about carrying 5 gallon buckets of water (at 8.33 pounds per gallon…) in 15 degree temperatures, or feeding in the dark before school and
mucking out in the dark after school; but unless you live south of the Mason-Dixon Line, this could be the reality for days or weeks on end.
Believe it or not, I’m not trying to discourage you or your child from horse ownership. Caring for a horse can instill a lifelong sense of responsibility and give your child focus and the wonderful feeling of caring for another living thing.
I’m just saying you need to be prepared for the reality. I got my first horse at the age of 12, and my family lived in New England. The days I described above could have been torn from the pages of my childhood – and I survived quite well! I even went on to own and operate a large horse farm in Connecticut for over a decade. (Although we did sell the farm and move to Virginia a dozen years ago – the cold seems harder to tolerate as I get older…)
My point here is that you need to have a plan in place if you decide to get your child a horse. Even if you keep your new horse at a boarding farm, there will still be the need for your child to visit the horse and ride regularly. Which means you’re going to be driving to the barn in the winter regularly. Are you ready to deal with sitting in a cold barn (or warm car in the parking area) for 2 hours while your son or daughter grooms the horse, tacks up, takes a lesson, cools the horse down, tops up water buckets… oh, maybe 2 1/2 or 3 hours…. 
My advice? If your child is a dedicated sort, by all means, consider horse ownership – but make sure you have warm clothes and lots of good reading material. Oh, and thank a mail carrier the next time you see one!