How much would an unexpected hash collision undermine the security of a hash function? Say some random person manages to, by sheer luck, find 2 messages with the same hash. There is no systematic way of creating more such pairs of messages that produce collisions.

How much would this one pair of collision producing messages affect the security of the hash function?
In many cases the collision might be exploitable to create many other collisions.  In almost all common constructions you could use the collision to build a collision of messages with arbitrary suffix. 
Building a useable collision for most attacks like colliding pdfs requires something about the prefix.
But even without a known attack on the relevant prefix I would be worried a different exploitation scenario exists.
Also I would be very worried about such "luck" which may suggest either:
a. The chance of collision is much higher than it seems
b. There is an efficient way of generating collisions it's simply not public.

Furthermore the structure of the collision may give important cryptographic insights.

So for a random collision I don't know of an exploitable attack. I worry there is such, I worry the collision could be extended, the crypo is weaker than it seems and may have already been broken. I would migrate away.