My Grandma's (RIP) name was Dorothy. Most her friends called her Dot.
Dill...ummm... I got nothing.
Dill could be short for Dillon or Dillard. Don't see those names nowadays. Offhand the only one I remember is Dillon Thomas, the Welsh poet. Oh, yeah, the construction guy from Chattanooga named Dillard Smith of Dillard Smith Construction company.