I think no one should get a horse right away before riding for a couple years on school horses first, especially if they have no experience prior or any professionals in the family. Riding a lot of different horses helps make you a better rider. I remember when I first started riding, I didn't want to ride any other horse besides the one I was already riding. Then I started moving barns and doing IEA, which I get a chance to ride many different horses as well. Now, I am proud to say I've probably ridden 25+ horses total since I first started riding 7 years ago, and now I finally have my own horse. Im not saying u have to ride for 7 years before u can get a horse, but just try it out to get a feel for it and try to ride as many horses as possible to get experience. Horses are a huge responsibility and there are many hidden costs that u might not have thought of before. It's probably best to take lessons for several months to see if your mom actually enjoys it and is willing to spend lots of money and even more time.
I would definitely recommend going to a riding centre first. Getting a horse is a big responsibility and no matter how experienced the horse is, your mother will still need to learn and understand riding. This is what I did and it definitely helps getting lessons first because then you can bring what you know and practice on your horse :)
I think you're better off learning to ride in a centre first, and then buying a horse. You never know whether or not your mother will truly enjoy it, or if the horse will suit. Then you're stuck with a horse that may or may not sell quickly afterwards. At least in a centre, you know the horses are suitable for beginners and your mother can find out if riding is something she definitely wants to do.