Well, they had the Iliad and Odyssey, which contains some information about the pre-collapse period (YMMV - depends who you ask. Personally, I suggest not asking Barry Strauss). They would have had Herodotus, and he claims (and I believe him) to be reporting tradition he learned from Egyptian priests. Hieroglyphs were apparently still understood by the priests but the knowledge to read them wasn't shared.

The Greeks had some stories which are tempting to put in context of the Bronze Age Collapse - there's the return of the Heracleidae (used to be connected to "the Dorian invasion" when people thought the Dorian invasion was a thing) and the invasion into Asia Minor of the Cimmerians. Unfortunately linking the Heracleidae to the Dorian invasion doesn't get us far, because aside from the literary tradition there doesn't seem to be good reason to assume there was a Dorian invasion, and dating for the Cimmerians is wrong by hundreds of years according to Assyrian records. You don't get very far if you're seriously trying to make these connections.

Here is a somewhat similar question about the Ancient Greek understanding of the Bronze Age collapse.