"one of the cables that runs" or "one of the cables that run"?

Hmm. I have just reversed myself on this, to the extent that I have removed my comment disagreeing with Chris M's answer.

Logically, it should be "run", unless it means that only one of the cables does run between the rooms, and we're choosing that one.

With the plural it parses as "one of [the cables [that run between rooms]]", giving what I think is the more likely meaning.

If you use "runs", it should parse as "[one of the cables] [that runs between rooms]", which analytically only makes sense with the interpretation I mentioned above.

But looking in the GloWBe corpus, "one of the [plural noun] that [3sg verb]" occurs more than three times as often as "one of the [plural noun] that [base verb]", which suggests that my reasoning does not in fact describe the current language.

I had a look in the COHA corpus (Corpus of Historical American English), and found that the form with the singular verb hardly appears before 1880, whereas the form with the plural has instances going back to the 1840's; but overall, the singular form is more prevalent, though only by about 20%.

The thing that convinced me, though, was noticing the phrase "one of the things that bothers" in the results. I realised that "one of the things that bothers me" is completely natural for me, where "one of the things that bother me" isn't.

So I conclude that "runs" is in fact idiomatic in English, though "run" is possible. I'm struggling to account for it: clearly "one of the cables" is felt to be the subject of "runs"; but I don't understand where the relative clause fits grammatically.

When 'One of' is the subject of a statement/clause, it should be followed by a determiner and a plural noun and the verb should agree with the subject in number, i.e singular.

E.g: One of my friends is a pilot.

If 'one of' comes in a sentence with a relative clause, the verbs in the main clause and relative clause should agree with their respective subjects.

E.g: One of the cables that run between rooms is dangerously live.

More details and examples are here from the Practical English Usage by Michael Swan:

The correct from is the singular because you're referring to only one of the cables