Both of 2sentences are correct Japanese. ここに なまえを かいてください。 There is a paper or computer screen and you are requested to write your name on it. ここで なまえを かいてください。 Maybe you are entering a kind of gate. You are requested to write your name there to enter the place.

koko ni namae kaite kudasai:)

Koko de namae wo kaite kudasai is better. because this "de" is about activity.