The most prominent is an inscription donated to the museum in 1895, probably from Amathus (or Amathous) in Cyprus, and tentatively dated to between 500 and 300 BCE. It can be seen on this blogpost, alongside a number of other images from the collection (the other inscription in that post is in Phoenician). The language of the Amathus inscription is known as Eteocypriot; unfortunately, while the entire syllabary is known and understood, the language is not, so that while the language can be transcribed into the Roman script, it can't be read for meaning. The Cypriot script seems to be descended from the Cypro-Minoan script (another syllabary), which in turn derived from Linear A, an undeciphered and thoroughly mysterious script from Bronze Age Crete.
The text should be transcribed like this, according to Cyrus Gordon (unfortunately Gordon was a bit of a crackpot, but I don't doubt the transcription - it matches what I worked out from my sketch using online sources, especially ancientscripts.com, although I think Gordon's word divisions are based on the presupposition that the Eteocypriot language was Afroasiatic):
The tablet was bilingual with Greek (as it is of such a late date, this isn't surprising), and as such it can be translated as follows:
1: a-na ma-to-ri u-mi-e-s[a]-i mu-ku-la-i la-sa-na a-ri-si-to-no-se a-ra-to-wa-na-ka-so-ko-o-se 2: ke-ra-ke-re-tu-lo-se ta-ka-na-[?-?]-so-ti a-lo ka-i-li-po-ti
The polis of Amathusians, to Ariston (son of) Aristonax, nobleman.
Eteocypriot is not an Indo-European language. Neither is the Eteocretan language, for that matter. I doubt that they are Afroasiatic, either. You'll note that the script used is a syllabary, and marks vowels. The scripts of almost all Afroasiatic-speaking groups (except in Ethiopia) have separate signs only for the consonants, as Afroasiatic languages use consonant roots to create meaning. There is normally no pressure to record vowels accurately in Afroasiatic languages. Still, best to keep an open mind, and it might still make sense for the Eteocretan and Eteocypriot languages to be Afroasiatic.
Cyprus and Crete were first colonised around 10,500 BCE, by foragers (if I remember correctly), but fairly soon after them (around 7000 BCE) came a wave of farmers from southwest Asia, probably Anatolia, who brought sheep, cattle, goats, and wheat (including bread wheat). This was part of a wave of expansion from southwest Asia that brought Neolithic technologies, including agriculture, pastoralism, and ground-stone axes, into Europe and the Mediterranean. It has been claimed that this expansion matches with the spread of Indo-European languages, but it doesn't, and it doesn't make any sense that it should (and it certainly doesn't correlate with the eastward spread into Persia, Central Asia, and India, which shows similar diversity to the spread in Europe but which demonstrably occurred thousands of years after the diffusion of Neolithic traits into Europe).
The migrations into Crete and Cyprus are clearly distinct from the more famous migration into the plains of Thessaly, in Greece, although in other respects they represent the same diffusion of the same technologies. There is much less bread wheat in Thessalian remains than in the islands, and other stylistic devices are different, so they seem to be different migrations (they probably were migrations, by the way, rather than the simple diffusion of the Neolithic complex into an existing population, although of course the picture was probably quite complicated). This means that we can't say that the Neolithic Thessalians - who indeed seem to have introduced farming into southeastern Europe - didn't speak Indo-European languages, just because the earliest Cretan and Cypriot languages aren't Indo-European. They may have been part of the same spread of farming into the Mediterranean, in some sense, but that doesn't mean that they descended from the same population immediately prior to migration. What we can infer from this, however, is that the spread of agriculture doesn't have to be accompanied by the diffusion of a single large language family (as strong versions of the language/farming dispersal hypothesis contend).
I think it is reasonably likely that Eteocretan and Eteocypriot were spoken by the direct descendants of the first farming populations in the region. Whether their languages are related to any living languages is debatable - I doubt the Afroasiatic connection, and I can't think of any other possibilities. Even if they were related to the language of the Thessalians, which is possible but unverifiable, it is most parsimonious to believe that their languages were wiped out by other Indo-European languages that spread with a wagon-and-horse-riding-bronze-working complex coming out of the Pontic-Caspian steppe around the fourth millennium BCE, leaving no identifiable traces in any modern languages (see here about one plausible hypothesis for the relatives of these languages).
In any case, Europe and the Mediterranean region used to be much richer in both languages and scripts. The Eteo-Cypro-Cretan languages and scripts are clear and elegant evidence of this.