Writing in the Forward, Jay Michaelson argues that if J Street really wants to build momentum for progressive change in Israel, it should look at Israel's largely Sephardic/Mizrachi working class. These Jews, hailing predominantly from North Africa and the Middle East, lean considerably to the right, and bear a considerable amount of resentment towards Israel's Ashkenazi elite.
There is irony that this resentment ends up redounding to the benefit of the Israeli right, as non-Ashkenazi politicians have seemingly reached higher levels in Israel's more liberal parties (e.g., Amir Peretz in Labor or Shaul Mofaz in Kadima). Nonetheless, the Israeli peace camp is overwhelmingly associated with Ashkenazi politicians. And Jews hailing from the Arab world tend to be those with the most visceral disdain for Arabs (Beitar Jerusalem, whose fans recently went on an anti-Arab rampage through a shopping mall, draws its support overwhelmingly from the Mizrachi community).
Still, to the extent that a critical aspect of any pro-peace endeavor is building support for it on the ground, making inroads in this community is absolutely crucial. I noted this sort of left revitalization project as an alternative to Beinart's settlement boycott, and this only reemphasizes it. How does one gain the trust of a community that has been ignored for so long? Well, by listening, to begin with, and showing that one is responsive to their (legitimate) concerns. Attaching consideration for Jewish refugees in the Independence War is an obvious example of an issue area important for this community that has been repeatedly marginalized. I'm sure further engagement could come up with others (that, after all, is the point).