The Basque word for 'town/city' is hiri, with mute H. Could its origin be in Middle Eastern languages with its subsequent incorporation into Basque through the Armenian Kingdom of Cilicia?
There is also a Hebrew cognate for the word, which is עיר.
Are there any reliable sources of etymology of this Basque word?
This Basque etymological dictionary claims that the word is "of unknown origin", but that the variant forms (h)uri and iri have been recorded in toponymy of Rioja, Alava and Navarro (Basque-speaking regions) from the 11th Century onwards. This would suggest that they are of Basque origin (and would rule out your theory of incorporation through the Armenian Kingdom of Cilicia - as the forms were recorded in the region prior to the existence of the kingdom)
From a draft of Lawrence Trask’s Etymological Dictionary of Basque, with abbreviations expanded by me:
hiri (Labourdin, Bas-Navarre), hí(r)i (Souletin), iri (Alta Navarra, Salazar, Roncalese), uri (Vizcayan, Gipuzkoan) noun ‘town’. 1545 (but see below).
From *ili, of unknown origin, by rhotacism of /l/. Last form by Bizkaian /i/-backing. The Romans reported a town called Iliberris in south-central Spain, and this name looks remarkably like a Basque formation meaning ‘new town’ (berri ‘new’). But there is no other evidence for Basque speech so far south, and we must suppose either coincidence or an undocumented settlement in this area by Basque emigrants. A second Iliberris (modern Elna) is recorded in Rosselló (Pyrénées-Orientales) (Coromines 1995 s.v. Elna), [...] The form (h)uri is well recorded from the 11th century onward in the toponymy of Alava and the Rioja, while iri is equally well recorded from the 11th century in Navarra.
If you're willing to consider outright speculation, it's worth mentioning Sorin Paliga’s Urbian theory. According to this theory, there is a pre-Indo-European root *ol- / *ul- / *or- / *ur-, connected with the concept of a town, present in this Basque word, Latin urbs, the name of the Swiss city of Uri, the name of the Iraqi city of Erbil, Greek labyrinthos and several other words.
