An onion is a biennial plant which means it will make seed in it's second year or when it senses that it is in it's second year. You can leave them in the ground and let them make seed for your next years planting. Onion seed does not stay viable for long so you would need to plant them the next year. An onion makes a pretty flower when it blooms and brings in a lot of pollinating insects, so while your onion could still serve a purpose it will not be for eating. It might be possible (maybe) to use it as a green onion, but it will most likely send up a scape and flower very quickly.

Even in it's first year, if an onion plant is too large when you plant it (larger than the thickness of a pencil), and the weather turns cold, then warm again, it 'thinks' that it has been through a winter and will bolt.

Carol already covered the topic thoroughly.

When I have onions that remained in the ground over the winter, I leave them there because they really aren't going to produce a bulb suitable for eating. They bloom and the flowers are pretty and attract many tiny beneficial insects that also will act as predatory insects and pollinators in your garden. That is why I always leave a few onions in the ground each summer in order to have the flowers in early spring for the beneficial insects.

I planted some leftover onions underneath our pecan tree (outside the garden) one year and left them alone there and they stayed there, bloomed in the spring, self-sowed seeds and survived in that manner for several years before drought finally must have gotten them in 2011. I never harvested them or even really paid attention to them, other than to notice that the little tiny beneficial insects were buzzing around them when they bloomed.

If you want perennial onions that you can grow and divide, there are several types such as potato onions, walking onions and multiplier onions that do that. They only produce small bulbs though, not big ones like the onions we commonly grow for eating. They don't take the place of large-bulbed onions. I use those perennial onions more like shallots in cooking.