Fame
If you're assigned to a big enough case, then yeah—you could definitely achieve some degree of fame (or notoriety). Consider Robert Shapiro and Johnnie Cochran, both now household names because of their defense of O.J. Simpson during his murder trial. (Ask your parents.)
Fame can make your career, and it can break it. It really depends on how good a job you did when you were being nationally-televised, and how seedy the man or woman that you were defending was. You don't really want to become known as the guy who got the clearly lunatic woman off the charges of driving her three kids into a lake.