Sola,
First, you assume that a person must have a reason to take it off rather than they just want to. Having the title "Baptist" in your name is not only not required Biblically, I do not think you would need any reason.
Secondly, you asked what bad things do people associate with the word Baptist? I could list a myriad of things and I hope others could. We have not always had a good name.
1. Perceived racism of Baptists (especially Southern Baptists). As you know, I am a Hispanic and have witnessed racism among Baptists and this was a stumbling block to me when I was a young Christian (and before I was a Christian). While I know not all Baptists are racists, I have had to battle this in churches for years.
2. Their hardshell beliefs and stands. A number of books have been written by many who survived these camps or whose life was severely altered in these camps. There were real people hurt by many in these camps.
3. I think theologically is another reason people would change. As I noted before, I would get rid of the name to avoid having people say we are lying. I am reformed, non-dispensational, and many have perceptions based out of a 20th Century view of Baptists which is not the historic view of Baptists.
4. The liberal Baptists. I do not want to associate with the likes of people who embrace many of these beliefs in the liberal realm.
5. The social stands/political many of these groups take which do not advance the Gospel but is an embarrassment. Do we really want people to associate us with people who say, "Don't vote wrong, vote Right." Or if I am ministering to other Hispanics, do I really want to associate with people who are hardshell in their political immigration policy? Or, do I really want to make a big issue on the Trans-Continental Highway? I am very political, but that has always been completely separate from my church. Yes, we should stand up for clear issues of marriage and for life, but when the Baptist largest organization boycotts Disney, I say, "you gotta be kiddin me."
6. Do I really want people to associate me with all the following people: Jimmy Carter, Tony Campolo, Jerry Falwell, Rick Warren, The American Baptist Churches USA, the Baptist World Alliance, Molly Marshall, Cooperative Baptist Fellowship, Al Gore, Bill Clinton, Fred Phelps, Jesse Jackson, Roy Moore, and more. Not really!
My question to you, tell me what Baptists means that unites everyone? Charles Spurgeon in the late 1800's said that it essentially has become defined of only believer's baptism, nothing more. I agree that in today's realm the word "Baptist" is mainly useless. In the South, it may have a better definition, but what most would equate with "Baptists" in the South (hypocrites who have little more than a social club and political fest) is something I do not want to be associated.
So, why would you want the name Baptist? What real meaning does it have anymore, that is unless you are a theologian and study church history.