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C'est vrai que c'est une jolie pierre, mais ça ne valait pas le coup de s'exciter à ce point, non ?
It seems that the noun coup tends to be used to express a strong emotion, but I’m not sure exactly how to interpret this phrase.
Also, does this sentence still make sense if you leave out “le coup de”?
Is the “coup” in this context perhaps an informal expression that means something along the lines of “thing/matter/situation”? Is there a similarity between “le coup de” and “la peine de”?
"Coup" is not to be considered alone in this context. As Begueradj explained in their comments, it is part of an idiom:
Litterally: to be worth it. You'd have the same trouble explaining what "it" and what "le coup" are. They vaguely design what you'd have to do, to invest to reach the objective, but as often with idioms, explaining individual terms is difficult.
The comments also mentioned "en valoir la peine", which was very rightly translated to "be worth the trouble". To be fair, though, the English version with "trouble" is mostly used (in my knowledge) when it actually involves the trouble. While your example mentioned "Paris vaut la peine d'être visité au moins une fois dans sa vie." in which case, though the trouble may be really limited, the word is absolutely not out of place.
It even sounds better, because "valoir le coup" is a bit vague and seems to be more of an oral than written expression.
You can also find "valoir le détour" (litterally again: "to be worth the detour/the trip", though it can also be used in a more figurative way, but often when the thing being worth is the opportunity to see something exceptional).
You might be confused by the fact "coup" is also used but with a narrower meaning in English which translates to coup d'état while in French, this word has a much larger range of meanings.
From the original/etymological one "hit, slap" to anything that occur more or less abruptly coup de poing (punch), de feu (gunshot), de téléphone (phone call), de balai (sweep), coup de main (give a hand), coup d'état (coup), coup de grâce, boire un coup (have a drink), ...*
Here, ça ne valait pas le coup is just a familiar way to say ça ne valait pas la peine (it didn't worth the trouble).
Ça ne valait pas cannot go on its own. You have to add a noun to the sentence.
For example :
