"See you ON next Monday" may be the most traditionally grammatically correct answer, but I have never ever heard anyone use that phrase. "See you next Monday" is the way all native English speakers I know phrase it.

See you... (when?) ...next Monday. (I assume you say this on the current Monday.) ...on Monday. If you mean the Monday following the coming Monday (ie. over a week away), use "Monday week". This means "the Monday after the Monday coming". That is, if it's Saturday and I say "See you on Monday", then I mean I will see you in two days. If I say "See you Monday week", then I mean I'll see you in nine days.

I would add that 'Next,' when discussing a week day is tricky even for English speakers. It can cause confusion. If we will see the person on the Monday literally following the spoken sentence, the sentence would likely be "See you Monday." If you throw next in, the person listening may not know if you will be coming Monday, unless it has been discussed you will not be seeing each other till the following Monday. Confused? Good. "I will see you next Monday" is often followed by: "Will I see you Monday- or anytime next week- or not?" Next is loaded with confusion- because in this situation, next means 'the one following the next.' The first next is silent!

"See you next Monday" or "See you on Monday" are the best responses.

You just say 'See you on Monday." Next Monday confuses things. As Jesse wrote it means the following Monday and not the one coming.