Sounds like hydrogen sulfide as noted by Larry. You have a couple options; an air-making system should work. If you already have one and it's not working, make sure it it properly pressurized with air (if your power goes off and you run the water, the tank will lose air pressure).
Another option is a bleach injection system to chlorinate your water. That's what we have, and it works great. There is a 15 gal tank next to the well tank that has a bleach/water mixture (4 oz of 12% bleach per gallon of water). An injection pump squirts small amounts of the bleach mixture into the water line exiting the well tank whenever there is flow, triggered by a flow switch. It's simple and works well. About $400-600 to setup, plus labor (but easy enough that a DIY'er can do it).
In our case, we were living in our new house for about 2 months before the smell started becoming noticeable, and only from some faucets and the fridge water. It took a good 7-10 days to drive the smell out of the fridge after we installed the bleach system, but it was finally gone and has not returned.
By chlorinating the water, you kill the bacteria that causes the hydrogen sulfide. The bacteria itself is harmless other than causing the smell. The amount of chlorination added by this system is much lower than typical public water supplies, by the way. So the water still tastes good, and only has a very light chlorination smell. Not nearly as bad a the "city" water lots of folks live with, which smells like a swimming pool!
I think if you talk to the guy who drilled your well, he will know what to check and will have some good suggestions.