A rum-based Manhattan is a Palmetto. A white Palmetto, with blanc vermouth, unaged cane juice rum, and tiki bitters is a nice summertime variation—we have one on the menu at our rooftop bar.
I have a Pusser's-based Brooklyn variation floating somewhere around Kindred. I don't think rum Brooklyns (or Greenpoints or Cobble Hills or Bushwicks or...) are made or called for enough to have common names, but they can be very tasty (as well they should be, featuring rum).