Rabbi Yehudah Hechassid wrote in his testament (Testament of R. Yehudah Hechassid, sec. 23) that a man should not marry a woman whose name is the same as his mother’s, and a woman should not marry a man whose name is the same as her father’s. Some authorities say that this testament was only intended for Rabbi Yehudah Hechassid’s own descendants, and others argue that this is good advice for everyone (Piskei Dinim Tzemach Tzedek, Yoreh De’ah 116). How do people hold these days? Is a common name with a parent a deal-breaker, or is there a way around it, such as a name-change?
Rabbi JD Bleich is strongly opposed to the name-change business, it can create bona-fide halachic problems. Suppose Shprintza wants to get married, but her mother-in-law-elect is also named Shprintza. So we go through a name-change ceremony, she's not "Shprintza" anymore, she's "Gendendel." Great. A few months or years later, people feel they've fulfilled their religious obligation and everyone starts calling her Shprintza again. If G-d forbid she'd ever need a Get, this becomes quite messy.
Basically, in some Hassidic communities it's taken seriously; in Modern Orthodox circles it's ignored; for Litvish Yeshivish it probably depends on your circles. Someone asked Rabbi Moshe Feinstein about it -- "may I marry so-and-so, but what about her mother's name..." Rabbi Feinstein replied -- stuff like this, if it seriously bothers you, then you have to keep it; otherwise, you don't. If you're writing to me for a heter, that proves it doesn't seriously bother you, so go for it!