What is a good substitute for onions when it's called for in recipes? I recently discovered that I get headaches from onions.

I would hate to loose my daily onion in cooking.
Consider discussing with your doctor 'and' reputable nutritionist. You might be able to totally stop using for a period of time, then s-l-o-w-l-y reintroduce into your diet...especially as cooked and not raw.
Worth exploring - good luck with it.

In India to add a garlic-y flavor and maybe sort of onion-y they use asafoetida. It's more of a powdered resin and is EXTREMELY powerful. A tiny bit goes a long way so 1/8 of a teaspoon is often more than enough in a normal pot of beans, or the tomato sauce, etc. It is available in international and Indian markets. You may want to give it a try, but I have no idea if the compound that gives the flavor is similar, so be cautious. Best luck to you.

My mom has trouble with aliums too. If sweating vegetables for the base of a dish where you want to caramelize for depth of flavor, try celery, carrots, kohlrabi, apples, or potatoes (maybe a waxy kind to avoid excess starch). I read once that onions, apples, and potatoes have the same sugar content and merely different textures--once cooked down, they really do resemble each other in flavor.

That is truly unfortunate for you, but here are a couple of ideas. I have a personal chef client who is on a low Fodmap diet. Garlic and onions are on the list of avoids. Her issue is IBS and other stomach issues, so not positive this will work for you with your symptoms. If it's a serious enough issue to be under a doctor's care, be sure to check with him/her first. Apparently, what causes the problems in FODMAP folks in onions and garlic are water soluble, but not oil soluble. Here is a link to an explanation by a FODMAP expert. If you scroll down in the comments, you'll see onions are included.
http://blog.katescarlata.com/2012/01/21/garlic-infused-oil/

Also, maybe this spice will work for you.
http://www.thekitchn.com/from-the-spice-cupboard-asafoe-140001

OH, you may want to try leeks before you entirely cut it out. Before my allergy got so severe I could substitute leeks for onions. Not anymore.

I'm allergic to alliums so I don't use onions or garlic at all in anything. While there is no direct substitute I use celery, celery seed, fennel, and lots of herbs. There's something called savory that is fantastic. I use a lot of cumin - LOTS. Try mustard and mustard seed. Lemon too. You will wind up changing your palette entirely but it's not something I miss at all. People who stay at our house or come for dinner have never noticed there were no onions or garlic in any meal.