You should check with the regulatory body for lawyers in the jurisdiction where you want to work. Ours at least has rules that you have to practise under your legal name. A good friend of mine got married young and legally changed her last name to her husband's. However, she still used her pre-marriage last name socially and when she started law school, she decided to use it because her husband is a lawyer and she didn't want people to assume she was getting special favours. She used it all through law school, made all her connections as a law student with that name, was involved with various organization, had a couple of awards etc. and then when she went to register with the law society, was told she had to use her legal last name, so that's what she uses now and people often don't realize that Jenny Smith and Jenny Thompson are the same person. You can use a middle name as your first name as long as you register with the full name, and a lot of people who do that are M. Brian Ducharme, or whatever. And you can't switch names around - they need to know who you are and every client has to be dealing with the same person. But every jurisdiction is different.

If your concern is being associated with the "privilege" of your dad, half the people in law school have parents who are lawyers or judges - at least mine did, and if not in the legal profession, then in other prestigious professions. I have never seen it be a negative.

Also, are you someone who thinks you would change your name if you get married? Because then you'd have to go through all the hassle and expense twice.

My kids have two last names in the Hispanic tradition. They're still young, but I haven't had any issues filling out forms for them yet and their schools/day cares haven't seemed to have any issues with it.
Get used to me. Black, confident, cocky; my name, not yours; my religion, not yours; my goals, my own; get used to me. -Muhammad Ali