A.N. Jannaris in section 148b and note 1 of his Historical Greek Grammar lists 'ὑγίεια, ὑγία or ὑγεῖα' and cites as evidence 'in inscriptions the proper names are invariably spelt Ὑγεῖα Ὑγεῖνος (twenty-four different names; never Ὑγιει- ; cp. the Index to CIA iii p. 746). Moeris cites the first as Attic, the second as Hellenic.
We can go a stage further (without the precision possible at this point), with thanks to the LGPN, but its presentation has its limitations.
All spellings of the feminine personal name are lemmatized s.v. Ὑγίεια (60x) and all of the masculine personal name, by contrast, s.v. Ὑγῖνος (154x). Of the reported spellings of the latter, there are many -γεῖ-, some -γιεῖ-, a few -γῖ-, and many Latin spellings Hyginus. Few are Hellenistic, many are Imperial Roman, and a few are Byzantine.
Of the former, -γεία- and -γία- (often as Latin Hygia) prevail. The majority are Roman Imperial, with one each for Hellenistic and Byzantine individuals.
GVI 646.4 (Sparta, II Jh. n. Chr., but BC in IG V, 1, 726) gives a metrically guaranteed example of Ὑγίεια. Ionic in dialect, but note Ἀλεξά̄νωρ.