I thoroughly enjoyed Alec MacGillis's "defense" of the New Hampshire primary, which was mainly an excuse for him to tell some good stories about First in the Nation events past and the Granite State in general. Good fun!
As far as New Hampshire, and for that matter Iowa...the argument for them is pretty basic. First, if we're to have primaries and caucuses, then (I would argue) a sequential system is better than a national primary, because it maximizes the influence of party actors and reduces the risks of accidents; in Polsby's terms, it increases the incentives for coalition-style candidates and reduces the chances of factional candidates winning. Second, because the key players here are (national and local) party actors, it's not all that important which states go first. Third, a stable schedule from cycle to cycle helps party actors maximize their influence. Fourth, there's at least a reasonable argument for the one-on-one campaigning available in small and mid-sized states. Therefore, given that Iowa and New Hampshire have been first for a long time and that both are small or mid-sized, we should keep them in place.
Since the modern system solidified in the 1980s, I think it's unlikely that a different sequence would have yielded different nominees in either party. It's probably the case that it would have changed the story of the nomination battle...it's possible to imagine John Glenn instead of Gary Hart losing to Walter Mondale in 1984, or Phil Gramm winning early before losing to Bob Dole in 1996, or George W. Bush clobbering someone other than John McCain in 2000. Or some forgotten candidates, a Bruce Babbitt or Pete DuPont, doing what Hart did in 1984 or McCain did in 2000. Would Clinton/Obama/Edwards have turned out differently had, oh, West Virginia gone first? I suppose with a contest that close it's certainly possible, but I pretty much doubt it. I do think that 1976 and 1972 could have produced different nominees on the Democratic side had the sequence been different, but that's mainly because I think the results in both of those years was basically random.
And that would be my main concern about significant change; it has the downside risk of producing random nominees until everyone is equally adept at working the new system. Remember, by "everyone" I'm especially concerned here with party actors. Their coordination problems are serious, and with an unstable system (as we had in 1972 and 1976) I think the chances of either coordination failure or a decision to withdraw from attempting to coordinate are disturbingly high.