I'm with Backinthebox. Not just a bad idea but potentially a very dangerous idea for your mum. A 4 year old is very very young, absolutely everything you do with him will be a learning experience and most youngsters will go through a 'terrible teenager' phase between 4 and 6 when they will try it on. This horse is very likely to shatter your mum's confidence, leave him to someone who has the time, patience and experience to bring on a youngster.
Why not get her a been there, done that hack who is getting on a bit? most horses' prices drop after 12 years old so you could get yourself an older horse that knows its job, is happy to be ridden once or twice a week and will look after your mum.
I think it's a terrible idea. There are so many nice older horses out there that have had a good education that would probably tick off all requirements. A 4 year old, however nice, needs educating. Schooling twice a week is just that - schooling twice a week. It's not a full education. It needs someone who will be able to hack him out sympathetically, being calm and strong if you meet something it hasn't seen before, such as tractors, cows on the other side of the hedge, barking dogs, etc.
For example, what if your novice mum was on him by herself for a hack after riding him with you for a few months and suddenly felt she was brave enough to go out alone, only to find it was bin collection day, and she was stalked slowly round the village by a piece of large equipment that rattled with recycling bottles every few minutes and smelled funny? Would she have the ability to reassure him, and not let him take off with her? You can say he is nice and quiet, but there WILL be something that will make him nervous at some point.
My own horse is 6, I've had him since he was 3. He is a really nice, easy horse. (I have 34 years riding experience and still consider myself to have lots to learn.) His education has included schooling sessions, being ridden by a variety of people, hacking out in as many different situations as possible - in company (where he is expected to lead the ride or go behind politely at all paces from walk through to galloping across open fields) and alone, fun rides, shows, hunting, riding club rallies, jumping lessons, and more. He still has a long way to go, even though he is a very nice person and easy to do. (I recently took him to his first dressage outing, where he surprised us all by behaving like a complete lunatic! More dressage training for him, then!) We have a 3 yo at home too, who is intended for a child, but will have his main education done by an experienced adult with the child hacking him occasionally, and the child also has the use of a nice 14 year old pony too, for her to build up more experience on.
FWIW, when my moderately experienced mum showed an interest in riding again after a few years break, I got her a 13 year old schoolmaster - not a completely novice ride, but a known quantity, his education complete, and sure of himself in most situations. He had been out and about and seen a lot of life. She was able to really enjoy her hacking without me or her worrying about what the horse might do.
My daughter who is 7 has a 3 yr old. She has groundwork lessons with him and they are getting along just fine. She hops on his back now and then for a trundle. Obviously a slightly different scenario for you but no reason why it can't. Just a bit of patience required. Good luck.
My mum wants to get back into riding, I have 2 horses but they aren't suitable to learn on (fizzy/ screw loose and too small for her) my friend is selling a horse who I have seen is very quiet, good to do, the right size.
However, he's only 4. If I schooled him twice a week to bring on his education do you think he will be okay for my mum to hack on? She will be hacking with me. Probably on a lead rein until she gets her confidence. And I will be about to get on if any bad habits start? Or is it a terrible idea?