Here’s a quote from a Stack Exchange answer to this question:
“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!”
For those thinking of following this custom of R’ Yehudah the Chossid, have you read the rest of his testament, and his book Sefer Chassidim? Do you follow all his other advice? Then why follow this of all things?