Given how abhorrent racism is, charging that Santorum used the n-word is not to be done lightly, so it's worth taking some effort to figure out what happened.
My interpretive transcription of what Santorum actually said is below.
"We know, we know the candidate Barack Obama, what he was like -- the anti-war, government-n...guh, the f--uh, the uh...America was a source for division around the world...."
So what is the word in bold?
One approach to this question is to play fill-in-the-blank.
If we guess the n-word, the result doesn't make much sense. In fact it doesn't look like a noun belongs in that spot at all. It looks whatever the phrase "government-n..." turns out to be is intended to act as modifier for a noun to come later.
Now, by Google, we know that Santorum's verbal repertoire includes a modifying phrase that fits -- "government knows best."
Let's plug it in:
We know, we know the candidate Barack Obama, what he was like -- the anti-war, government-knows-best....
Looks to me like a fit grammatically and contextually.
So my guess is that he meant to say "Barack Obama, what he was like -- the anti-war, government-knows-best...." but then could not figure which noun to use to bring it to a close, so he choked off "government-kn..." such that it came out sounding like "nig," then set out in another rhetorical direction with "America was a source...."
Plausible?
12:24 PM PT: UPDATE: based on a couple of comments, it seems I need to make clearer what I think happened. I think Santorum cut himself off after the "n" sound in what would have been "government knows best."
Commentors are pointing out that what Santorum said doesn't have an "o" sound, but an "i" sound. I think that's true, but I also think it sounds plausibly like the sound you would make if you cut yourself off right after making the initial "n" sound in "knows."