watashi no namae to (anata wo mitsumeru)shisen (or manazasi) wo wasure nai de kudasai. 私の名前と(あなたを見つめる)視線（まなざし）を忘れないで下さい。 adds (anata wo mitumeru) , it will be natural japanese.

Watashi no me to name o wasure de kudasai. Jaxchan's "metsuki" probably works better in this case (it refers more to the expression of the eyes). "Wasure de kudasai" is a somewhat polite form of don't forget ("wasure de kudasaimasen ka" would be more polite). I don't know if "wasurenakute" (the negative te-form) would work here as a plain negative imperative command.

Watashi no namae to me wasurenaide kudasai.

:) You don't say such things in Japanese. ;)

Watashi no metsuki to namae wo wasurenai yo.