All urban legends. Being raised Italian, with all of the onions that we used, I would be dead by now and not turning 66. With properly covering the onion and refrigeration, you can eat onions left over in the refrigerator for at least a week, if not more.
If you are really worried about it, chop up the rest of the onion, put it in a Ziploc and freeze it. You can then put the leftovers in casseroles, stews, soups or anything that will cook the onion really well.