Unfortunately we have no hard evidence one way or another, because Homer uses both, and that's the oldest Greek we have. (Mycenaean inscriptions sometimes help us go farther back, but they're no help here.)
However, we can make a very good guess. There's a well-attested Ancient Greek verb lanthánō, "to escape notice"; the extra n comes from a present-tense marker inherited from Proto-Indo-European (it comes from *lh₂-n-dʰ-, from *l-h₂- "to be hidden"). In non-present tenses, you see the stem lēth- (*).
Since this verb clearly comes from a PIE verb (it has things like the nasal infix that weren't productive in Greek), and there isn't evidence of the river Lethe in other Indo-European mythologies, it's most likely that the verb lanthánō came first, the noun lěthē came from the verb, and the name Lěthē came from the noun.
EDIT: The caron on ě indicates a rising tone, which is written with an acute accent in the original Greek. (e ē é ě ê = ε η έ ή ῆ)
(*) This is called the "nasal infix", and it's one of the ways Proto-Indo-European indicated the present tense: sticking an n in the middle of the word. It died out in every Indo-European language I know of, but you can see fossils of it: for example, English victim and invincible both come from the Latin word for "conquer", but invincible came from the present (and thus had an n) while victim came from the past (and had no n).
There's one and only one instance of n in a native English word, rather than a borrowing. That one instance is present-tense stand next to past-tense stood.