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.
٢٠ ديسمبر ٢٠١٨
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koko ni namae kaite kudasai:)
٢٠ ديسمبر ٢٠١٨
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Koko de namae wo kaite kudasai is better. because this "de" is about activity.