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I will allow you to perform an essential task for me, one that many of my followers would give their right hands to perform....
From Harry Potter and the Goblet of Fire.
The closest phrase I can find is "give a hand" meaning "help someone". But I'm not sure what "give their right hands to" means exactly.
The general idea being expressed is that other followers are so devoted to the speaker that they would be willing to make a great sacrifice in order to perform a task for him. "Give up their right hands" is an example of a great sacrifice.
From the context of this quote alone, we would assume that it is simply metaphorical. I.e. the speaker is merely symbolically describing the followers' devotion, but there won't actually be any right hands literally given up. However, at the end of the book it turns out that
(warning: spoiler ahead)
the "essential task" involved the actual physical removal of the right hand, so the expression ended up referring to a literal sacrifice, not just a figurative one.
It looks like a variation of the idiom give an arm and a leg:
give an arm and a leg (for something)
- To pay, give, or do whatever is necessary or required. They had to give an arm and a leg for the lawyer they hired. I had to give an arm and a leg just to get an interview for the job.
- If you say that you would give an arm and a leg for something or to do something, you mean that you want it very much. I'd give an arm and a leg for a cup of tea. Any jazz professional would give an arm and a leg for half an hour playing with Wynton Marsalis.
(TFD)
In other words, other followers really want to do those tasks, or they would do anything necessary to be able to do those tasks.
Apparently, this is from Harry Potter and the Goblet of Fire. According to @Alex (*spoiler alert*), the statement foreshadows actual hand sacrifices.
In other contexts, other variations could be made, like "give your right leg". A crude example I've heard would be "I'd give my left/right nut for [something]", nut being slang for testicle.
As I understand, it is a way of saying pay the ultimate price. In this case in order to be allowed to get to do something. In general most people are right handed, and losing a right hand would be a very high price to pay for something. There also could be some link with historical concept of the right hand being considered to be the clean hand in some earlier cultures. Cultures that existed before the invention of toilet paper, in which the left hand was used for that purpose and therefore not used for any "clean" thing. A person being forced to use the left hand to eat and other tasks would then be an outcast from society. In which case, a person who gave up their right hand would not only have a difficult existence, but would be totally cut off from their own people.
The phrase "Give their right hands to..." is an idiomatic expression which implies that a person has given their due consent as regarding a decision.
