Since my childhood, I have known hundreds of various religious, both men and women. I even live very close to a Benedictine Monastery of monks, where I did my high school.
Do religious choose their own religious name?
First of all, it depends on the particular Order or Congregation, as some institutions still keep the tradition of their religious changing their names, while others do not. This applies to both men and women.
Amongst those places, where this tradition is still in place, whether of men or women, the aspirant gives three names to their immediate superior to choose from. The superior is free to choose from these three names, but is not bound to choose from them either. It is in no way, the individual Religious who makes the final decision in their new name.