Constitution is vitally important. More durability for all classes (essentially a bigger health bar), and as a spellcaster, a better ability to maintain concentration on spells. There are other benefits but those are the biggest ones.
However, dumping con is an incredible role play opportunity. You could equate having low con with a frail or infirm person - someone who gets sick easily, can't eat exotic food and spices, or gets drunk off of one drink. It could be a child adventurer. It could be someone with a disease who is on a quest to find a cure, or a magical item that increases Constitution (I'm drawing a blank on the name of it atm).
Your player dumping con is giving you the DM an incredible sub plot to go along with your campaign.
Edit: wrote "in" and corrected it to "I'm".