It is a "display of confidence", meaning "submit myself to something unpleasant if I prove to be wrong" as in:
If the train arrives on time, I'll eat my hat.
To say "afterwards I had to eat my hat" is to admit that you were wrong.
http://www.phrases.org.uk/meanings/eat-my-hat.html
As was pointed out on that other question you linked, this is a surprisingly tricky sentence!
Let's slowly build it up.
It will take someone else.
What is "it"? Stopping Voldemort again. Implied but never directly said. As for "take", we could say "need" instead, just to be slightly clearer.
[Stopping Voldemort again] will [need] someone else.
This isn't "someone else" as in "we need a different person", but as in "we need other people prepared to do what you did, Harry". When will they be needed? "Next time" (that is, the next time that Voldemort tries to return).
[Stopping Voldemort again] will [need] someone else who is prepared to [do the same as Harry] next time.
Now, Dumbledore is saying that this someone else doesn't have to do much. Hence, it will "merely" (or "only") take someone else. This may sound like it's demeaning Harry's efforts, but it's meant to be reassuring Harry: standing up to Voldemort is not very difficult or unlikely after all, and it doesn't take someone extra-special to do it.
[Stopping Voldemort again] will merely [need] someone else who is prepared to [do the same as Harry] next time.
What, exactly, did Harry do that Dumbledore says they need other people to do (or be prepared to do)? "Fight a losing battle"--or what seems like a losing battle. (Dumbledore omits the word "like", but I'll leave it in for this one example.)
[Stopping Voldemort again] will merely [need] someone else who is prepared to fight a losing battle next time.
[Stopping Voldemort again] will merely [need] someone else who is prepared to fight what seems [like] a losing battle next time.
And now we just replace the bracketed bits with the different wording Dumbledore uses, and we have the sentence (okay, part of a sentence) that you bolded.
It will merely take someone else who is prepared to fight what seems a losing battle next time.
Best Answer
To "put someone in control" means to give them the ability to make decisions that govern the behavior of something.
From context, what users are being put in control of is document processing, and so, presumably, they have control over one or more printers and/or photocopiers.
The sentence does not make clear precisely how the users are controlling document processing. Perhaps it might have to do with scheduling jobs on a slower printer or photocopier with a lower cost-per-page, or setting up a number of printers in a priority queue, so that jobs are routed to the least expensive printer available, or so that drafts are printed with a lower print quality requiring less ink.