There are always three alternates within fuel range plus 20% for holding. They plan for that when they fuel the plane. The one exception is trans-ocean flights. If they get near the %20 threshold, they can declare emergency and clear the runway and approach for immediate landing. Watch
this if you want an understanding of how it works. At 1:53, they declare emergency. At that point, the ATC clears the pattern, hold all ground traffic short of runway 22L, rolls ground emergency crews, and proceed through the checklist for declared emergency. They coordinate incoming aircraft for landing since this particular emergency does not involve low fuel. Emergencies like this happen every day around the world. The general public never hears about any of it because the vast majority of them end with no issues due to the incredible amount of redundancy and caution in commercial flights.
As far as Erie closing their runway(s?), they most likely made the call after your mom's flight was in the air. When the intended destination became unavailable, they chose one of their 3 alternates. Detroit was the best option.