*h₂ēiti 'he shares'From this, one might then take *h₂éisos as the basis for the pan-Italic word *áisos 'god' (Venetic aisus 'god', Marrucinian aisos 'to gods', Paelignian aisis 'to gods'), which in turn goes back to *h₂ei- 'to share'.
→ *h₂ēist 'he is in a state of sharing' [sigmatic stative] → 'he reveres'
→ *h₂ēis 'respect, honour' [athematic root noun]
→ *h₂éisos ~ *h₂éisis 'honour; honoured one' [thematic noun]
→ *h₂éisḱeti 'he begs' (<? *h₂éisǵeti, intensive form of *h₂ēist)
→ *h₂éisdet 'he gives respect' (short for *h₂ēis deh₃t ?)
If this idea flies right, then the commonly held view that Etruscan had imprinted their own native term onto the Italic languages would be in the wrong direction. It's very easy afterall to pin a word origin to an obscure language and play the 'out of sight, out of mind' game. So some skepticism of the status quo is healthy here.
The Eteo-Cypriot form *aisona (a-so-na), which in my view probably means 'god-offering' as it does in Etruscan, would have been borrowed from Etruscan sometime around the Orientalizing Period (c. 8th century BCE). The Italic loan implied by Etruscan ais 'god, goddess' then would have been transferred even earlier, say, right at the beginning of the Etrusco-Rhaetic immigration into Italy: the 10th to 9th century BCE.
This relates to Andras Zeke's latest thoughtful entry, The Kafkania Pebble - testament to the strangest of religious practices?, speaking on the artifact's legitimacy and possible links between a-so-na written on it (as in Eteo-Cypriot) with Etruscan aisna 'divine (adj.); god-offering (n.)'. If the word is an Italic loan though, this word on the pebble must be something else and/or the artifact could be fake. The idea that this artifact dates to the 17th century BCE strains credulity in my eyes. Nonetheless, this particular word and its origin is fascinating to delve into.