Prof. Howard Friedman (Religion Clause) has details, including links to the complaint and its Appendices. Newdow -- who filed the lawsuit against the use of "under God" in the Pledge of Allegiance -- is seeking the removal of "so help me God" from the oath to be read by Chief Justice Roberts, and exclusion of the clergy invocation and benediction. (Note that Newdow is not seeking to prohibit President-Elect Obama from saying "so help me God" in his oath.)

The lawsuit's Establishment Clause argument about the inaugural prayers is foreclosed by Marsh v. Chambers (1983), which held that legislative prayers are generally constitutionally permissible, even to the extent they may endorse religion, because of the long tradition of such prayers dating back to the same Congress that proposed the Establishment Clause.

I can't speak with equal confidence about Newdow's argument that the court should at least enjoin any prayer that focuses on a particular denomination, rather than just being generically Judeo-Christian monotheism (itself a denomination, but one that Marsh suggested was capacious enough for government work). See here for a hint of the debates in lower courts about when and whether denominationally specific prayers are constitutional, though there are many more cases on the subject than just the one I mention there. I should note, though, that Newdow's argument on this has been rejected before, in the decision rejecting his lawsuit about the 2005 inauguration. And a President's inviting a particular clergyman to say things at the President's inauguration might well be treated as an extension of the President's own right to express whatever views -- including denominationally specific views -- he wants to express as part of his own speech.

The Establishment Clause argument about the "so help me God" in the oath is likely also foreclosed by Marsh, given the long tradition of "so help me God" in oaths. (Oaths, after all, were supposed to be invocations of God, as opposed to affirmations, which were the constitutionally prescribed alternative for those who didn't want to swear to God.) There is of course a dispute about whether President Washington said "so help me God" in his oath. But it's clear that early oaths -- including the one for federal judges and Justices, plus several other examples from early Congresses -- commonly contained the phrase. The logic of Marsh would thus amply apply here.

The lawsuit's Free Exercise Clause and Religious Freedom Restoration Act arguments are foreclosed by the requirement that the government action must "substantially burden" the claimant's religious practice. Under the caselaw that has developed as to substantial burden, being offended at the government's use of religious language in a government ceremony would not qualify.

There is also the possibility that Newdow can't relitigate the matter now, because he had filed similar lawsuits over the 2001 and 2005 inaugurations; but I don't think this would bar the other plaintiffs. The plaintiffs might also lack standing to litigate this, but I doubt it, given that some of the plaintiffs claim they may be present at the inauguration. In any case, I leave those procedural questions aside here; people who are interested in them might want to read this decision about the 2005 inauguration.

So reports Religion Clause (Prof. Howard Friedman), pointing to this decision.