Any difference between "give up something" and "give up on something"
In general, giving up something means simply leaving or abandoning it, while giving up on something means to stop trying or pursuing it. So the general flavor of the construction is that "on" makes it a bigger deal.
In this example, though, I don't see a lot of practical difference. If there is a nuance, I would attempt to characterize it like this:
- I have given up the idea of going to college.
Sentence 1 seems to imply a more simple go/no-go decision. Giving up the idea of going could imply a "singular" or limited scope, and "the idea of" could seem like superfluous words. So the interpretation would be essentially "I have given up going to college". This sentence conjures an image that perhaps you were casually toying with going to college and decided not to.
- I have given up on the idea of going to college.
Sentence 2 could reflect more philosophical cogitations. The "on" implies that what it refers to is substantive. Giving up on going to college would be like the first example. Giving up on the idea makes the idea seem more substantive than in the first example.
The image this sentence conjures is, for example, maybe a long family history and expectations of college attendance, or a prolonged consideration of college as part of your life plan. So the second example represents an extended body of thought about college attendance, all of which is being put to bed.
I may be reading too much into it, but that's sort of the meaning of nuance.