I'll go first.
An apparatus can be made to fire a single photon ("light particle") and a detector can detect a single photon. So photons were thought to be particles by a lot of folks.
But then there's the Double Slit Experiment, where light was shone through one slit which was followed by a barrier with two slits, and photons ended up in places that were totally crazy, not in straight lines through the slits like reason would tell you they would have to be. Like this:

So that pretty much proved that light particles were only particles sometimes, but other times they were not particles at all but were rather waves that could interfere with each other like how sound waves can cancel out due to phase relationships.
It doesn't really make any sense, that you can shoot one photon through a slit and then two other slits, and it might end up in any number of places, including places that it couldn't actually get to if it was travelling along a path, but hey, there ya go!
Somebody please explain this better if I got it wrong.
I would love to hear other examples of modern, cutting-edge science coming to conclusions that sound totally absurd. Please.