Replace or repair derailleur?

Replacing it is the more economical option in almost all cases, especially on lower end stuff (esp. once you include labor) -- you can't source the parts for many lower end derailleurs.

On higher end parts it may be possible to send it out to a company which rebuilds derailleurs or rebuild it yourself with parts, but again, the labor and parts will likely outweigh the cost of just getting a new one (and the new one is less likely to fail; the rebuilder may make screw ups).

So, I'd almost always replace, unless I had a good reason not to (e.g. someone who insists on vintage parts and wants that vintage part repaired). Or if its a tiny deviation (which wouldn't occur if the derailleur went into the wheel; that's generally toast).

I would rather have the mechanic bend it back and offer a money back guarantee if its not perfect. Cheap and no risk to me as the customer. If you cannot (or are not prepared to) offer that guarantee, don't do it. Even if you are prepared to take the risk the customer comes back, the customer still has to come back, so won't be entirely happy with you.

If you do what the customer asks, and it does not work, it costs the customer more. Even if he understands it was his choice and won't blame you in any way, how do you know this? How do you know the customer will not 'rant off' at you, or worse, your other customers about you, for doing a 'crap' job? Whats your situation with consumer protection law - will he come back at you and say the repair was not fit for purpose and demand his money back?

If you stick to your guns, you might loose a job, but really, is it a job or customer you want to keep.

This answer plays off a bit of ambiguity in the question. Below, the emphasis is mine.

If you were a customer would you rather have the derailleur replaced and know it is fixed, OR have a mechanic attempt to repair it by bending it back or replacing the cage and accept the problem may not be fixed the first time?

The derailleur cage is the two pieces of metal (or sometimes carbon) that hold the derailleur jockey wheels. It's below the parallelogram of the derailleur. I'm quite sure that cage is standard terminology. By parallelogram, I mean the main body of the derailleur, the one where the logo is usually printed. I'm not sure if that is the universally accepted term.

The cage plates are a replaceable item. The site guidelines prefer us not to link to specific online retailers as the links may not be persistent, but currently, I'm seeing one online retailer (Modern Bike) asking for $32 for both plates for a Shimano Ultegra R8000 rear derailleur. Dura Ace (made of carbon) and many Campagnolo cages (many made of carbon + Campagnolo premium) are much more expensive. Alternatively, people sometimes sell damaged derailleurs for spare parts on eBay. A crash is likely to scrape the face plate (or whatever the correct term is), but damaging the cage itself is less likely. Therefore, cyclists should probably consider checking the price of replacement parts before replacing the entire derailleur.

The cage on my Shimano Ultegra RD does not respond to a magnet, and I believe it's aluminum. In general, it seems not recommended to bend aluminum back to its original shape, as it may later fail. If the cage in question is steel, bending it back would be much safer, but there may be issues aligning it correctly, as you have no obvious reference points for how it's supposed to look.