I will go out on a limb and suggest that the Supremes will rule it's a state issue, not a federal one. I base this on the fact that, outside of the EC meeting, there is no federal election of the president, it's all done at the state level.
I would be very surprised if they demand proof of eligibility. While there is a constitutional eligibility test, the document does not say who has the authority to check up on it and do something. The tenth amendment likewise gives a hint of what direction they are likely to take.
While this is all very frightening, it's pretty entertaining too.
It certainly IS a federal case...President of the US is a federal position!
What they rule for one state is a precedent for all the states in any case...most of the states will take the same action as NJ. What SCOTUS rules applies to all states.
BTW, there is also a lawsuit brought on by the one for VP of the Independence Party of the US and also one brought on by 24 of the electorates of the EC. By this time, there may be more electorates joining the latter case. They are refusing to give their electoral votes for Obama unless he has first shown his eligibility.
Whatever happens, it is already a constitutional crisis. Hope O goes to prison for a long time if it comes to that!