ILLYRIA

1. In ancient times the country used to be written as Ἰλλυρία, so, it should be pronounced as [illooria], not [illiria] (the problem of ypsilon).

2. The suffix -ur is Kartvelian and is used to form adjectives out of nouns.

3. In Georgia exists a surname Iluri, derived from a noun 'ili'.

4. The double ll in the name of the toponym can be explained as follows: 

We know that Kartvelians don't like the doubled consonants. At the same time, they practice the omitting of vowels while forming the words: megobAro -> megobroba, mgEli -> mglis, etc. So, if a vowel is omitted between the same consonants, the doubling is inevitable.

There is a Kartvelian verb ილულავს [ilulavs] meaning "half-closes one’s sth (eyes, from drowsiness, bright light)" (Rayfield). The root here is ilul. If we try to form out of this root an adjective with the help of -ur suffix, we'll end up with iluluri, which has very high chances of being reduced to illuri.

Have-closing of eyes or squinting is a typical feature of many nations in Asia and Europe. For example, take a look at a classical Illyrian Gellert Ivancsics, a Hungarian football player of Yugoslavian origin: