What does the word "rich" mean in the reactionary sarcastic phrase "That's rich!"?

Solution 1:

According to the Online Etymology Dictionary, the sense of "entertaining, amusing" is recorded from 1760, so it's been around for a long, long time.

Source.

Solution 2:

I'm familiar with the word being used in the phrase "That's a bit rich coming from you", here's a definition I found:

http://idioms.thefreedictionary.com/That%27s+rich!

That's (a bit) rich!

something that you say when someone criticizes you to show that you do not think they are being fair because they are as bad as you. "I'm greedy? That's a bit rich, coming from you!"

Cambridge Idioms Dictionary, 2nd ed.

Solution 3:

UK national here, entirely speculating but perhaps it refers to something tasting rich, but used ironically so as to make it synonymous with distasteful.

So it might be used to mark offence that something said was harsh, or rude.

From experience it is most often used in this way in response to perceived hypocritical statements.