What a lovely thoughtful thing to do, to reach out for advice to help you support your mum!
I'm a few years older than your mum is, but I know what she means. Life starts to close in on you a little bit as you get older - I know we shouldn't let it but it does. Does your mum go out at all? Does she work?
I work FT, I have 3 adult children, some still at home or back and forth. I've also got good friends even if I don't see them all the time. I think I'd do more of that if I didn't work FT, as do all of them who live in the same country as me.
What does she enjoy? I love travelling, and you and she could go on wonderful trips abroad, if that's feasible financially. I have had three lovely trips away in the last three months but have nothing planned so that's a bit of a downer lol!
She should really contact her GP and arrange to see him/her face to face. It's hard getting through, and I hesitate too because it's such a pain in the backside. I don't see why they wouldn't see her. There's no shame in admitting that you need a bit of help and support and she should access that.
Do you still live at home? If not, do you think she is lonely? Does she rely on you too much though? That's not a good thing either really.
As for a cat or dog, you'd have to make sure she was happy to look after a pet and could financially and physically manage it. A cat would probably be easier - I have three indoor floofs. They are great company and they make me laugh.
You could go for walks, for coffee, to National Trust properties, to the beach (even in cold weather!), for dinner, to the cinema. No reason why you couldn't go shopping either - though if you are anything like my DS, who hates it, you're probably best not to!
Does she enjoy reading for instance? Sometimes I just love to snuggle up with a good novel and my own company.