Zeno's paradoxes were part of the reason the Greeks were so conflicted about anything involving infinities or measurements.

The Banach-Tarski paradox is a great example of why Axiom of Choice is controversial. You can essentially make two spheres out of just one sphere that are all the same size.

Not sure if there's a name for it since it's not really a paradox, but for a century, mathematicians couldn't easily explain why the definition of a derivative didn't involve dividing by 0. People knew that their current understanding of it seemed to involve dividing by 0, but they also knew calculus was right since we could apply to physics. It led to the development of analysis, which later provided more precise definitions for things like limits and derivatives that don't involve any division by 0.