I have a family member who is a US Citizen, but currently lives overseas. They want to open a US bank account and are asking me if I would be ok for them to use my home address for the account. In principle, I would be happy to do them a favor, but just want to check whether this could potentially get me into any legal trouble?
If it isn't legal, then I'm curious as to what law it would be breaking.
The main problem is that your family member is likely providing fraudulent information; you're merely a willing accessory. However, it's not automatically fraud; if the bank knows your address is merely a mailing address then it's not fraud.
The practical upshot is that you can give your family member permission only for the latter; it's legally his problem if he then misrepresents that to his bank. But you may still have problems with debt collectors in that case.
