The name Plummer is a boy's name of English origin.
Plummer might be an occupational name for someone who works with pipes -- yes, like a plumber -- or with feathers, from the Olde English (from the French) plume. Or it could indicate someone who lived near a plum tree.
The plum association is delicious; the plumber association, not so much. If you like occupational names, better go with Gardener or Sawyer.