Making a name for yourself means you have achieved some notoriety or recognition for your work or are associated with a particular trade or skill.

1. Gary left college and moved to Hollywood to try to make a name for himself as an actor.
2. I worked very hard to make a name for myself as an athlete in my town.

1. Sarah was very nervous when she first started, but quickly made a name for herself as one of the best teachers at the school.
2. The boss made it clear that employees who were more interested in making a name for themselves than being part of a team would not last long at the company.