mid-15c. (late 12c. as a surname), a variation or alteration of Middle English rother, from Old English roðor "paddle, oar," from Proto-Germanic *rothru- (source also of Old Frisian roðer, Middle Low German roder, Middle Dutch roeder, Dutch roer, Old High German ruodar, German Ruder "oar"), from *ro- "steer" (from PIE root *ere- "to row") + suffix *-þra, used to form neutral names of tools.
The original sense is obsolete. The meaning "broad, flat piece of wood attached to the stern of a boat and guided by a tiller for use in steering" is from c. 1300. For shift of -th- to -d- compare burden (n.1), murder (n.); simultaneous but opposite movement turned -d- to -th- in father (n.), etc.