I'll outline a few of the reasons why Ben Witherington has offered the view that the audience is predominantly Jewish.
1:17 If you invoke as Father the one who judges all people impartially according to their deeds, live in reverent fear during the time of your exile.
2:9 But you are a chosen race, a royal priesthood, a holy nation, God’s own people, in order that you may proclaim the mighty acts of him who called you out of darkness into his marvelous light.
Witherington sees these as suggesting a Jewish audience, since they would naturally refer to Jews and not to Gentiles. For when are Gentiles ever referred to as exiles in their own regions? [Witherington, 2007; 28]. Furthermore, Witherington asserts that vs. 9 "is a direct echo of the Petateuch's report of what God said to Israel." [pg. 28]
On one of the two decisive passages in 1 Peter that appear to suggest a Gentile audience, Witherington suggests, regarding 2:10 that:
First Peter 2:10 is frequently seen to be a clear proof that the audience must be Gentiles. Here we have an intertextual echo or partial quotation of Hosea 1:9-10. Could our author really have been referring to Jews by phrases like “once you were not a people” or “once you had received no mercy”? This in some ways is a very odd question when one reads the original text of Hosea in its own context, where Hosea is clearly speaking of and about Jews, and offering a prophetic critique of their behavior. The prophet is indeed talking about Israel being temporally rejected and then restored. Thus there is no good reason why the author of 1 Peter could not be using this language in the same way as some of his own Jewish contemporaries. The key perhaps is to recognize that our author, himself a Jew, reflects the view of over-Hellenized Diaspora Jews that was not uncommon among more Torah-true Jews, who had been raised and lived in a more conservative environment in the Holy Land. For instance, consider the reaction of Qumranic Jews to Hellenized Jews in Jerusalem and elsewhere. [28-29]
Witherington's got a point. But is this enough to establish that the audience is Jewish?