My school did not allow PNM's to make their own name tags. For first round, you wore a Panhel-issued standard name tag. For subsequent rounds, sororities made name tags for the PNM's they'd invited back.
We had a girl come through rush one year, who had a nickname that she greatly preferred to her given name - so she altered her Panhel name tag accordingly. We took note, and made sure that the name tags we made for her had her nickname on them. It must have worked... she joined AEPhi

(BTW, if you do make your own name tag and you have a nickname, make sure that the name you used to register for rush appears somewhere on your name tag... you don't want sororities not inviting you back because they couldn't find your name on the list of PNM's.)
We had another girl come through rush (different year) - she had been invited to my sorority and one other for second round. As I said, each sorority gives you a name tag when you arrive; when you leave, you're supposed to remove your tag and either keep or discard it, since the next sorority you visit will give you a new tag. Anyway, this girl wore her name tag from the other sorority right through our party, next to the name tag we'd given her... as if to say "if you want me you'd better rush me hard, otherwise I'm going XYZ".

She wound up not being invited to pref by either sorority.