make a name
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)
Fig. to become famous. Sally wants to work hard and make a name for herself. It's hard to make a name without a lot of talent and hard work.