While saying "I didn't have little fun last night" would not be correct, saying "I had no little amount of fun last night" would. Why?
There is a weird figure of speech in English called litotes (lie' toe tease) which uses a double negative in creative ways. For example, in keeping with your "lots of fun" quantifier, with litotes you could say,
We had no little amount of fun last night.
Translation: We had lots of fun last night. By negating lots, you get little. Then by negating little, you imply lots!
Just to reinforce this strange figure of litotes, consider the following two sentences. Which one contains litotes?
There was a huge crowd at the party last night.
There was no small crowd at last night's party.
Sentence #2 used litotes.
Is there any particular reason a person would choose to use litotes instead of the more normal way of expressing a thought? I suggest two reasons. First, litotes introduces a variety in the way a thought would normally be expressed. Variety is good. For this reason, however, its use should not be overdone, or then the ab-norm becomes the norm.
Second, by negating the opposite of what would normally be said, folks with an ironic bent may use litotes to satisfy a certain desire within them to express things ironically, for irony, too, delights in the conflict of negatives, as when the literal is negated by the figurative.
Speaking ironically about last night's party, you could say to a person who simply loves interpersonal conflict,
I heard there were lots of fights, plenty of heated arguments, and no small number of insults and barbs traded back and forth at the party last night. You must have had a blast!