I don't think dictionary definitions would help much here, so let me create some example sentences and try to explain the nuances of each word choice that way.
That's bizarre.
This sentence means that something very surprising, shocking, and out-of-the-ordinary occurred.
That's eccentric.
He's eccentric.
"That's eccentric" doesn't sound right. Eccentric is usually used to describe people. "He's eccentric" means that the person behaves very oddly.
That's funny.
That's funny. I would not have expected to find my keys under my couch! Maybe the dog dragged them there?
That's funny. Earlier you said that you hadn't seen Jill all day. But now you're saying that you saw her at the store right next to where the crime occurred. I think you may be lying!
The second sentence has a nuance meaning that the speaker did not expect whatever happened to happen. It conveys puzzlement about the current situation.
Funny can also have a slightly negative connotation. It can be used to accuse people of lying. An example is provided in the third sentence.
That's odd.
Same as "that's funny". The nuance is that the speaker did not expect whatever happened to happen.
That's strange.
Same as "that's funny". The nuance is that the speaker did not expect whatever happened to happen.
That's unusual.
Slightly higher register than "that's funny", "that's odd", and "that's strange". It sounds like a well-reasoned, carefully considered statement rather than a passionate, honest, in-the-moment comment.
That's weird.
That's weird. I would never wear such ugly socks. People would laugh at me.
Same as "that's funny". The nuance is that the speaker did not expect whatever happened to happen.
This word may also have a slightly negative connotation, like in the second sentence. It can convey disapproval of a behavior.
Hope it helps! :)