Which is correct?
I will see her on the weekend.
I will see her during the weekend.
Neither. The answer is “this weekend”, as in “I will see her this weekend.” Depending on which weekend you mean, you could also say “next weekend”, which is the weekend following “this weekend”.
“On the weekend” is sometimes used, but sounds odd to me. “During the weekend” would only be applicable if you were clarifying that you meant not before or after, but during the weekend.
It's pointless trying to make a case for any difference in meaning between any of these prepositions. The main area where usage varies is that Americans favour "on the weekend"
whereas Britons favour "at the weekend"
Both nations also use over the weekend (for Americans, this is actually the most common form; for Brits, it's a close second). But for Brits, during the weekend is the least favoured of these top four prepositions (it's second-to-bottom for Americans, who really don't like at the weekend). And I'm pleased to report that in the weekend just isn't in the running according to NGrams!
Most speakers will be quite safe with this/next weekend, because most people in any particular linguistic community will have a common understanding of what these terms mean, but it's worth knowing that others may think differently, particularly when you're online and interacting with people who don't necessarily share your local idiomatic usage.
On the weekend is the normal usage. You can use during the weekend if your meaning specifically involves periods of time rather than just sometime "on the weekend."
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