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.
As someone commented, I work during the summer could mean a couple of things. It could mean that one works only during the summer. Perhaps it is a student who takes a job, and then explains to his friend:
Yeah, I don't have time to relax during summer break. I work during the summer.
However, this phrase may also take on the meaning that one works during the summer, as well as during winter, fall, and spring.
On the other hand I work during summer usually only implies that one is already working, and keeps working during the summer period. One example of this use might be a summer school teacher. He might say:
My students during the school year are great, but I work during summer to make ends meet.
Interestingly, during the summer is more common than during summer according to this NGram:
However, you could pretty naturally say either phrase in order to imply what you mean. In context, the listener will figure out whether you mean "summer only" or "including summer".
Best Answer
"During" = concurrent with but not necessarily a component of the process
"In" = a component of the process