Many pronouns have this kind of genitive form
Genitives in -ius exist for a fairly small number of Latin words. I'm not sure of the exact amount.
I would say that the stems that take this kind of genitive form constitute a "closed class", although there are a number of derived pronouns that inherit the inflection pattern of the base pronoun (e.g. quisque, cuiusque built on quis, cuius).
The usual pronunciation seems to be with long -ī-, as -īus, although some forms in short -ǐus are also attested in poetry, and in eius, huius, and cuius the -i- is assumed to be a long consonant [j.j].
This genitive ending is characteristic to words used as pronouns or as quantifiers/determiners that are (otherwise) declined with first/second declension endings. So despite being uncommon, its usage can be argued to be predictable and not entirely arbitrary. It could be considered irregular if you wish. It goes along with the use of -ī in the dative singular.
There are a few pronouns with complications in the genitive singular:
The pronoun nēmō is derived from the third-declension noun homo. Its genitive is not formed as *nēminīus. A third-declension genitive in -is, nēminis, is occasionally found, but not in general use: generally the suppletive nūllīus from nūllus is used instead.
The pronoun nihilum has the second-declension genitive nihilī. No form *nihilīus seems to exist. However, I believe the use of nihilī is restricted in general to the idiomatic context of the "genitive of value" ("homo nihili"). The phrase "nūllīus reī" is attested in cases like "ex me audiebas mihi illum ex multis variisque sermonibus nullius rei" (Cicero).
I haven't put together a complete list yet.
- eius, huius, cuius
- illīus, ipsīus, istīus
Derived from these pronouns:
- eiusdem, cuiusdam, cuiuscumque, alicuius, cuiuspiam, cuiusvis
You can see some with quantifier/determininer uses listed in Allen and Greenough 113:
- The following nine adjectives with their compounds have the Genitive Singular in -īus and the Dative in -ī in all genders.
- alius [aliud (n.)] other
- tōtus whole
- alter, -terīus the other
- nūllus no, none
- ūllus any
- neuter, -trīus neither
- sōlus alone
- ūnus one
- uter, -trīus which (of two)
b. The genitive in -īus, dative in -ī, and neuter in -d are pronominal
in origin (cf. illīus, illī, illud, and § 146).
c. The i of the genitive ending -īus, though originally long, may be
made short in verse; this occurs often in alterius and regularly in
d. Instead of alīus, alterīus is commonly used, or in the possessive
sense the adjective aliēnus, belonging to another, another's.
The etymology of Latin genitive singular forms is complicated
The etymological origins of this genitive singular ending, as well as the regular -ī genitive singular ending of the Latin second declension, are not straightforward and so appear to have been extensively discussed in linguistic literature. The only paper I have read so far (and only partially digested) is "The o-stem genitive singular: Considerations from the perspective of the Latin dialects", by Luca Rigobianco (2017, Études de linguistique latine II).
Rigobianco rejects a common hypothesis that derives cuius, huius, eius from genitive singular forms in -osio- (cognate to Sanskrit -asya-) questioning the validity of the loss of -s- in this phonetic context (10). The references mentioned in favor of the hypothesis are Leumann, 1977, p. 477 and Meiser, 1998, p. 117.
I haven't read a source yet that directly discusses the forms with vocalic/syllabic -i- such as unius; it's my assumption that they would in some way be related to cuius, huius, eius.
The origin of the final -s is another difficult point; it has been explained as an added element by analogy with genitives in /s/ (as in nouns of the third declension or archaically in the first declension).