Origin: I don't give a rat's a$$

So as promised…not really to you dear reader, but for my own enjoyment, I looked up the origin to giving a rat’s ass and here are the top contenders (insert drum roll here – not literally. You need to use your imagination. Man oh man, has the computer world really created people who can’t use their imagination. Hear it now? that’s better)….

This is very likely to be one of many 'I don't give a...' variants. Animals get used for all sorts of things. Here, in the UK, it used to be 'as sick as a dog'. That's now out of date and the current phrase is 'sick as a parrot' - without, I may add, any evidence whatsoever that parrots are particularly sick. It's all 'Fashion'.
This is my least favorite explanation because well, the writer is from the UK and talks about Fashion…nuff said!

Have you forgotten that the hoot-owl's hoot is used in the same manner?"I don't give a hoot about the man in your moon." "I don't give a rat's ass about ..." See? Same idea. Owls get picked on, too.
Another animal one. Not bad but well, not good either.

It's just general slang that started up sometime during the 20th century. While nobody knows for sure how it came about, it's believed that a rat was used in the expression because of the way people view the rodent (annoying and small).
Better…more along the lines of what I’m thinking when I write it.

But the winner is…
When the term was first coined, presumably a rat's ass was the least valuable thing that the speaker could possibly imagine. If the subject of a sentence doesn't even "give a rat's ass" about something, the implication is that he or she certainly wouldn't give more valuable things, such as concern, or a second thought to the of the sentence.

This is what boredom with my life has reduced me too. Eeee gads!!!!