Labda (Λάβδα) was a daughter of the Bacchiad Amphion. More significantly, Labdacus (Λάβδακος) was a king of Thebes, a city whose early dynasty was named after him (‘Labdacids’), and whose son Laius begat the (in)famous Oedipus.
The Lydian equivalent of the Greek god Zeus (Ζεύς) was Lews (𐤩𐤤𐤥𐤮) or Lefs (𐤩𐤤𐤱𐤮), bringer of rain. It seems fairly obvious this deity’s name has been affected by the Anatolian ‘tl’ sound and so may derive from an earlier *tlews < *teos << IE *deywós.
I would observe then that the form Lefs is strikingly similar to Labd- in the above two names. Was Labdacus a local ‘Zeus’ and Labda a ‘Zea’?
[ Note 1 ]
The Lydian deity Lamẽtrus (𐤩𐤠𐤪𐤶𐤯𐤭𐤰𐤮) is equivalent to Demeter, equally obviously doing back to a *tlametr- < *tametr-.
My own favoured etymology of the name ‘Demeter’ is ‘house mother’ (ie ‘mistress of the house’), as per Despoina (Δέσποινα). These have a root in ‘dem-‘ (‘house’, ‘to build’), the Lydian equivalent to which is ‘tam-‘, a perfect match for *tametr-.
[ Note 2 ]
Looking into this a bit further, here are a few more outlier ideas on the matter.
First, the word labrys (> labyrinth). This is another seeming ‘tl’ word, for it can I think be found in the name of first Hittite king Labarnas/Dabarnas. Linear B has the form da-pu2-ri-to-jo-[po-ti-ni-ja] for ‘labyrinth’ and this could be pronounced ‘δαφυρίνθοιο (daphyrinthoio)’. This now reminds us of daphne (‘laurel’) and Latin laurus is thought to be related to it. This latter is close to Etruscan lar ‘lord’ which likely is related to labyr- and labar-.
Can this cluster of words by linked to labd-? Or *deywós > lef- / ?*leb-?
The name Labarnas I think means ‘chief’ or ‘king’. The word labyrinth would therefore have an obvious meaning of ‘palace’. Perhaps labrys is the root, for *labarn- would be ‘the one who holds the labrys’. Is the labrys the Latin laurus? The da-pu2-ri-to-jo the ‘place of the daphne‘?
Working back still further, is this the plant of the *tleb > *leb — the god? The bundle of laurel become the double axe of Minoan Crete?
Finally, if we have this word for ‘chief’ lar, labarn etc and if Labyrinth and place names such as Larissa mean ‘chief’s place’ or ‘palace’ or ‘fort’, is that the meaning of the ancient palace of Lerna? Is *lerna is dialectal variant of *labyrinth and *larissa?