According to Lifehacker, put onions in the freezer for 15 minutes before chopping them. It may make it slightly more difficult to cut and remove the outermost layer, but it can significantly reduce the irritation to your eyes. When chopping onions, tears can be induced because of a volatile sulphur compound in the onion. When it reaches your eyes it causes them to sting and tear. When you cut onions, that gas is released. If you freeze the onion it slows down the enzyme activity thus reducing the release of the compound.
On a personal note, I’ve found that if you start cutting at the opposite of the root end of the bulb it can help too. Bon appetit!