Larissa Bonfante informs us in Etruscan dress (2003), p.104 that, based on the Greek word τήβεννα (tēbenna), the toga was known to the Etruscans as *tepenna. While I gauge this to be essentially correct, it requires some ammendments to account for some minor, but important, facts that Bonfante has overlooked.
First, it must be known that Etruscan lacks geminate consonants. Putting it another way, Etruscan never distinguished between written -nn- or -n-, being pronounced exactly the same: plain old /n/. This is unlike languages like Italian where double letters in spelling can make a difference in the spoken language. Since it's unnecessary for Bonfante to be faithful to the Greek spelling with double nu (-νν-), *tepena would be a comparatively more sage reconstruction. However, a second interesting fact is that there are some dialects of Greek (like Doric) which preserve long a when other dialects have raised the vowel to long e (eta). So a meticulous linguist asks themselves whether they are to take the phonetics of this Greek word at face value or whether they should assume that eta reflects an earlier long a at the time this word entered Greek from the donor language (ie. tēbenna from earlier Greek *tābenna) . Adding these extra considerations into account, I believe that Etruscan *tapina is the most accurate reconstruction.
But now the fun's just starting. Earlier in Paleoglot: Minoan inscription HT 104, I began to realize that the Minoan word for 'carpet' or 'rug' must have been *tapia (note both Greek δάπις 'rug' and τάπης 'rug', strongly hinting at substrate influence). Through the lens of my personal view of a two-branched Aegean family, the adjacent Proto-Cyprian language should have shown *tapi by the end of the 2nd millennium BCE due to the regular syncope of all final vowels. This is precisely the form I would expect to be inherited into Etruscan too.
The exciting part is that, within the bounds of Etruscan grammar, a word like *tapi-na is a perfectly acceptable derivative of *tapi plus the very productive suffix of appurtenance, *-na. This can account for the term tēbenna morphologically and phonetically while it would just as aptly account for the attested semantics of the word as well! It might even be recommended (although not necessary to this theorized etymology) to tweak the value of the Aegean root *tapiya to a broader value of 'cloth' or 'textile' (ie. anything woven, not just rugs but also garments too). The fundamental meaning of the Etruscan tebenna then can be summed up as simply 'that from cloth'. This thought-experiment couldn't have produced a more mundane and sensible result.