What would the implications be of a single randomly discovered SHA-256 collision? Imagine if someone messing around with hashes accidentally came across two inputs to SHA-256 that produced the same output.  Not a concerted effort, not Google/NSA or crypanalysis. No tin foil hats.  Just random chance in say a high school. And they made the two inputs public knowledge.  Whilst highly improbable, not mathematically impossible.  And just that one collision.

What would be the implications?  Would people simply say "Well it had to happen someday" and carry on as normal, keeping the hash as is? Or would SHA-256 have been "broken" and an upgrade sought?

An answer to Are there any well-known examples of SHA-256 collisions? suggests that it would be major.  Even if it happen entirely by chance?
Your question is a contradiction. If it actually is randomly discovered, then there are no implications because it is, well, random. If it were truly the result of luck, then so what? But the more likely explanation is that SHA-256 is really, very badly broken. I personally would not believe that anyone came up with such a collision by chance without testing an enormous number of possibilities.

Note however that, due to the length extension attack, a single public collision could be used to make more collisions with the same prefix. This could have security implications in practice. Additionally, if the collision came about through a non-generic attack against the hash itself, it may be possible to adjust it to find more collisions (see Luis Casillas's answer). This isn't the case if the collision was random.

Think of it another way. If I posted a list of all the passwords you have ever used online, and claimed I just got lucky and was posting random numbers from an RNG, would you believe it, or would you instead suspect that I had hacked your computer and was stalking you?