Reading through some things, I came across the concept in both Rome and Ancient Greece of the Agnostos Theos or "Unknown God." From what I read, I got two different interpretations of the idea.
1.) "The unrevealed God". The idea essentially is that there were Gods that were not known to them that they still wanted to curry favor with. Therefore they would pray to them and offer sacrifices in a manner that could be described as "covering all their bases."
2.) "Whichever God handles this." Essentially that when you wanted to receive the blessing of the Gods, but didn't know which one to pray to, you prayed to The Unknown God in order to make sure that you at least tried to communicate with them.
The information I have found seems to point toward the first option mostly, but there is some hinting, at least in the Roman world, of the second concept.
The "unknown god" is known from only three independent sources. One of them is a Christian text, Acts 17.23:
[22] Then Paul stood in front of the Areopagus and said, “Athenians, I see how extremely religious you are in every way. [23] For as I went through the city and looked carefully at the objects of your worship, I found among them an altar with the inscription, ‘To an unknown god.’ What therefore you worship as unknown, this I proclaim to you.
Almost all other later sources are based on this passage (including a reference to "the unknown god" in the Philopatris spuriously attributed to Lucian, which is actually of Byzantine date).
The next is a non-fictional pagan source, Pausanias, Guide to Greece 1.1.4:
The Athenians have also another harbor, at Munychia, with a temple of Artemis of Munychia, and yet another at Phalerum, as I have already stated, and near it is a sanctuary of Demeter. Here there is also a temple of Athena Sciras, and one of Zeus some distance away, and altars of the gods named Unknown, and of heroes, and of the children of Theseus and Phalerus...
And the third source is a fictional pagan one, Philostratos' Life of Apollonios of Tyana 6.3, where Apollonios sees "altars of unknown divinities" in Athens.
All extant Attic inscriptions that use the root agnōst- ("unknown") refer to mortal individuals, mostly epitaphs; so there the word means either "illegitimate" or "of no tribal affiliation" or something of the kind. Nothing to do with any gods.
Between them, Pausanias and Philostratos show that it's not "the unknown god" (theos), as Luke/Paul would have it; it's "unknown gods" or "divinities", plural. In particular, Pausanias' phrasing
βωμοὶ δὲ θεῶν τε ὀνομαζομένων Ἀγνώστων
altars of the gods that are called the Agnōstoi
suggests that it's a particular group of gods, with "unknown" as an official title.
There are various philosophical and quasi-philosophical passages that equate any and every phenomenon in the universe to "the god" or to Zeus/Jupiter, e.g. Lucan's Civil war 9.580, where Cato defines Jupiter as "whatever you see, whatever motion you make". Or there's this passage that appears in a couple of Orphic Theogonies (Orphica frs. 14 and 31 ed. Bernabé):
Zeus came into being first, Zeus is the last, the bright-lightninged (god);
Zeus is head, Zeus is middle, all things are wrought from Zeus;
Zeus is the breath of all, Zeus the fate of all things fulfilled;
Zeus is king, Zeus is the governance of all things, the bright-lightninged (god).
That seems to be the kind of thing you have in mind; but there's nothing to suggest a connection to the altar observed by Paul, Pausanias, and Philostratos. There was a 1913 book by Eduard Norden, Agnostos theos: Untersuchungen zur Formengeschichte religiöser Reder (Leipzig: Teubner) which went into this metaphorical material and tried to equate it with Paul's theological treatment, but from the reviews that I've looked at he didn't make a very convincing job of it.
If you want to go into the "whichever god happens to deal with X" thing, I'd suggest steering clear of treatments that use the "unknown god(s)" terminology because it's so closely associated with Christian apologetics.