What is the correct tense of the verb "write" if I wanted to ask someone "Would you mind if I [write/wrote] your name on this list?"
To my ears it sounds like "Would you mind if I wrote..?" is more of an "in theory" question, while "Do you mind if I write ..?" is the way you would phrase when standing right next to a sign-up list with your mate and presenting him with an actual choice. Also I don't think it's correct to use "would" + present tense.
Should he confirm the latter question, you're GOING to write his name on the list. Asking "Would you mind if I wrote.." feels like asking someone if he'd generally object or agree to you writing his name on that list.
Anyway: Out of those two choices you gave, the correct one is "Would you mind if I wrote".
The use of write signals a likely event. The use of wrote signals a possible event.
Would you mind if I write your name on this list?...
...because that is what I plan on doing unless you tell me otherwise
Would you mind if I wrote your name on this list?...
...because I won't unless you tell me to