Over 60 lawsuits have been filed about Obama's eligibility and all have been dismissed. The issue is one of "standing." A court can't hear a case unless the court is entitled to hear the case, that there's been or will be a specific injury, and that the court can fix the problem.
Courts have held that it's a "political question" -- a power specifically delegated to one branch in the Constitution. They followed the Constitution, and have said that only Congress has the power to remove a sitting president. In addition, only Congress and the Electoral College can "qualify" the President-elect -- none of the Electors and no Rep or Senator raised an objection. Thus Obama was seated and sworn-in as President. So any lawsuit alleging that Obama was ineligible gets dismissed -- the courts have no jurisdiction (Congress does) and the Courts can't provide a fix (only Congress can).
So a lawsuit could be filed saying a law was unconstitutional (or that a serviceman's orders were given by an illegal Commander-in-chief) because Obama was ineligible. The lawsuit wouldn't make it past the Motion To Dismiss stage, for the reasons above on Standing and Jurisdiction. If the MTD was denied and the case went forward, then the defense (Obama's lawyers from the Dept. of Justice) would stay proceedings and appeal the denial. The case would never get to the point of discovery, let alone trial.
In the unlikely event that a case DID get past the MTD, then Obama simply gets a new copy of the Hawaii Certificate of Live Birth (posted on Factcheck.org) along with an affidavit from the Hawaiian Department of Health (Vital Records) that says how it was generated and that it matches the records they have on file. The Judge then would say OK, looks good to me, case closed.
Getting Obama's name on the ballot is a different matter. Objections can be filed in many states challenging Obama's qualifications to be listed as a candidate. Perhaps the Court of Public Opinion would require Obama to provide further proof. But in any event, birth in Hawaii is sufficient. Any challenge would require better proof than the State of Hawaii as provided --not just rumors.
Natural/Native born citizen requires birth on US soil. Parental status doesn't matter. That's been the rule in this country from the founding. There are only two kinds of citizens: natural born, and naturalized. There is no third class of "born in this country but not a natural born citizen": that was fixed by the 14th Amendment (slaves and their descendants were born in the US but couldn't be citizens.)
These three come with the highest accolades:
Lee A. Ciccarelli - 610-915-8446
Leonard Rivera - 610-840-0128
Charles Thomas - 877-397-3003