make a name for (oneself)
To establish oneself in a particular field or area; to become well-known. It took a long time and a bunch of supporting roles, but I've finally made a name for myself as an actor.
make a name for oneself
Achieve distinction, become prominent or well known, as in Martha is making a name for herself as an excellent chef. The earliest recorded use of this term was in John Wycliffe's followers' translation of the Bible (II Samuel 8:13): "Forsooth David made to him a name." Also see make one's mark.