My mother's grandparents were both born in the UK and moved to the US as young adults.
Can my mother apply for some kind of working visa or dual citizenship, which can be passed along to me so that I am eligible to live and work in the UK?
In general, British citizenship (or the equivalent status historically) can only be passed to one generation born abroad. Therefore, your mother would not have gotten British citizenship automatically, as she is in the second generation born abroad. Depending on her year of birth, it may have been possible to register your mother as a citizen as a child under certain conditions, but I am assuming that that never happened.
If your mother is a Commonwealth citizen, she may be able to get an ancestry visa to work in the UK.