Rabbits are lovely pets if you put in the time to make sure they're socialised well with humans and get plenty of attention to build up a proper bond. Yes, it's better if they live in pairs, but if you spend lots of time with them then they can live quite happily on their own.
We had a rabbit who was just lovely, more like a little dog than a rabbit, iyswim. We used to let him have the run of the garden and we would leave the back door open so he could come into the house to see us whenever he wanted. He loved being with us and would follow us around the garden, and sit with us for snuggles. He died a few years ago at a ripe old age and I still think of him fondly from time to time when something brings him to mind.
Rabbits aren't good pets for young children but they're great pets for older children. The problems start because people expect them to live happily while stuck in a hutch all the time without exercise and without attention from their owners. A little bit of effort will give them a lovely life and make them much better companions.
I think there is a special happiness that comes from having an animal friend, particularly for sensitive children. If it were me, I would get her a pair of rabbits from a rescue, or one rabbit with a view to lots of attention from her, every day. But I love animals anyway and would have a house full if I could.
I wouldn't recommend a hamster, tbh. They are SO loud at night, running on those hamster wheels, and they are definitely a useless pet in terms of what you get back from them. Not worth the effort or the money, IME. But I suspect that it depends on how much time you spend building up a relationship with them. I just couldn't handle all the noise at night, it used to keep my girls up. They don't live all that long, though, 18 mos - 2 years, I think, so at least it's not a long-term commitment.
If you want a good small pet, get a couple of rats. They are very cute, very smart and will really engage with you. Males are better than females, more docile and not quite as inquisitive and 'busy'. Females do make great pets, too, though.
If you hate the idea of having a pet, and will resent having to be ultimately responsible for it as the adult of the household, then of course I wouldn't recommend you get one. But if you're just unsure that it would add anything to your daughter's life and happiness, then maybe it's worth reconsidering as I believe it could greatly enhance her overall contentment.