Hey! That's a super interesting question actually, bc it brings about the meaning of modal verbs!
The sentence you wrote ''My mother could have given me some money if she'd wanted to, but she didn't'' means my mother had the ability to give me money, but she didn't want to.
Whereas the sentence with should ''My mother should have given me some money if she'd wanted to, but she didn't'' is a bit more emotionally charged and can be read as ironic. The speaker is essentially saying their mother never even wanted to give them money at all, and if she did, she simply should have.
You can *technically* use either one of the verbs, the sentence would be grammatically correct, but the meaning would be completely different, so at the end not the same sentence😆
You can look up meaning of modal verbs, or modal verbs meaning and read up on it! It's a super interesting topic, but it can be a bit confusing to people who just started learning. So if you already have a very good grasp on modal verbs and their basic usage, it might be worth taking a look :)
Have a lovely day⭐