Myself and Joel Willitts have thought a fair bit about whether Ioudaioi should be translated as "Jews" or "Judeans". See our respective posts on Judean and Syrian, Being Jew or Judean, and Its High Time to Change our Terminology. I'm not yet ready to jettison the term "Jew" in favour of "Judean", although I do think that "Judean" finds a proper place at many points especially related to the Gospels and in parts of Josephus. See also Phil Harland (e.g. here and here) and Loren Rosson's posts (here).
What has further raised doubts in my mind about treating Ioudaioi as "Judean" is an essay by Daniel R. Schwartz, '"Judean" or "Jew"? How Should We Translate Ioudaios in Josephus,' in Jewish Identity in the Greco-Roman World, eds. J. Frey, D. Schwartz, and S. Gripentrog (Leiden: Brill, 2007), 3-27. Schwartz notes now post-holocaust sensitivites have both helped and hindered discussions of Jewish identity and he notes the complex problem of identity as not all Jews adhere to Judaism and not all adherents of Judaism are Jews by birth, so life and usage can get complicated (a complication exhibited in Israeli courts where Jewish identity often has to be proven). He also acknowledges examples where Ioudaios clearly means "Judean" such as Apion 1.177ff and Antiquities 18.196. Yet Schwartz presents no less than ten reasons why we should prefer "Jews" over "Judeans"!
(1) Epigraphic evidence indicates that Ioudaios refers mostly to people who have been born as Jews, regardless of where they are from, and in a few cases to those who had converted to Judaism. (2) If the Idumeans, Judeans, and Galileans made a common front against the Romans, what is that front to be called? (3) In 2 Macc. 2.21, 8.1, and 14.38 Ioudaios defines a person by his relation to religion not by his place of Judea. (4) There seems to be no evidence at all for calling someone we would call a non-Jew a Ioudaios. (5) When we do hear of pagans mentioned in Judea they are usually called 'Greeks' not 'Judeans'. (6) Our English term 'Jew' refers not only to religion but also to descent, and much data in Josephus points to Ioudaios as something predicated by birth. (7) There is an element of development in Josephus' thought between Wars written in the 70s and his other works written in the 90s, Josephus' understanding of being Jewish developed from one which assumed that religion and state go together to one which recognized that they need not. (8) Greco-Roman authors very rarely linked the Ioudaioi with the land of Judea and they used other words for it such as Idumea, Palestine, or Syria. (9) Given that more Ioudaioi lived outside of Judea than in it (i.e. the Diaspora) there is not enough evidence to indicate that Ioudaioi could unambiguously be taken as linking those it denoted to a particular land. (10) There is no good reason not to treat Ioudaios just like Rhomaios. All "Romans" were Roman regardless of whether they were in or from Rome or not.
Interesting stuff to think about!