She wore round framed glasses with the left side slighter bigger than the right (one).
Do I need one in the following case? Why or why not?
I would say "She wore round framed glasses with the left side slighter bigger than the right side."
In the beginning you are talking about the side, so it would be better to end with side.
Unrelated to your question about "one": You should say "slightly" instead of "slighter".
She wore round framed glasses with the left side slightly bigger than the right (one).
On to your real question, I think using "side" instead of "one" would sound better. Using "one" sounds a little bit weird. (Although it's not too bad) It would make more sense to use "one" if you are referring to two separate objects.
If you decide to leave out side/one, it would sound perfectly fine. The word could be inferred from context. This would be called ellipsis.