'Could You' is considered to be an informal way of asking something, contrary, 'Would You' is a formal way of requesting someone to do something.

Can you vs Could you VS would you?

If we assume that the speaker has no reason to doubt the other person's ability to write their name, then "would" is the better word choice. The most proper way to use these words is to use "could" if you're not sure if the person is able to do something.

Could VS would with examples?

When I was in high school, I could do my homework on the bus. ... Doing my homework on the bus was something that I did do, not just something that I had the ability to do. Would is also a modal verb used to denote possibility, and it often refers to something that is conditional but definite within those conditions.