My 20-year-old daughter has a son. She’s a full-time student and I support both of them financially. Which one of us – me or my daughter – should claim my grandson as our dependent?

You should. Because you support both of them financially, you should claim them both as a dependent. Your daughter cannot claim her son as her dependent because she is a dependent herself. (26 C.F.R. § 1.36B-1(f); 26 C.F.R. § 1.152-1; IRS, Publication 929.)