Can a family member with a different blood type still give a kidney to me?

It Depends on Typing. Blood type is one of the first hurdles to identifying a candidate for organ donation. If your family member is type o, then there should not be a problem. If a -> you must be a or ab to receive; if b-> you must be b or ab; if they are ab then you too must be ab. Blood type is just one of many compatibility tests that are run. Hla typing and crossmatching are also performed prior to transplant.
No. To prevent rejection, your a, b, & rh blood-type antigens have to be compatible. A donor cannot have any blood-type antigen that you don't already have. In other words: ab is compatible with o, a, b and ab. A with a or o, b with b or o, and o only with o. Rh+ matches rh+ or rh-, rh- matches only rh-. It is often harder to find a match for the other major "histocompatibility antigens".