Well of course the answer is "we have absolutely no clue"...
...But I voted for the entire Middle East, because I do at least think it's plausible.
First, and perhaps most importantly, we have Iraq's huge neighbor Iran, which is teetering on the edge of a popular democratic revolution. When that will come, and how successful that will be, is impossible to say, although it seems likely that a successful democratic transition next door in Iraq would help things along (or perhaps make the ayatollahs more likely to agree to significant reforms short of an all-out overthrow of the regime). It's worth noting that as one of the most educated, sophisticated and cosmopolitan countries in the region, Iran has one of the best chances of a somewhat quick transition to real democracy. It's also worth noting that due to the fact that the government itself has been virulently anti-Western and anti-American in particular for almost 25 years, Iran has one of the most pro-Western, pro-American populations in the region. (The countries where the US is hated are precisely those in which the despotic regime is friendly with the US, worst of all those countries--especially Saudi Arabia and Egypt--where the regime has been to some extent propped up by the US.)
Next there are a couple smaller Gulf states where real democratic reforms are ongoing (Qatar in particular); these are likely to speed up or at least remain on pace.
Then Saudi Arabia. Now Saudi Arabia has started some very minor reforms, and talked up further ones, especially post-9/11. Whether this will amount to anything on its own is hard to say. A democratic, pro-US Iraq could help a lot by giving the US an easy way to lessen our involvement in SA and support of the Saudi regime; we will likely move most of our Gulf region military bases out of SA and into Iraq, and will hopefully be able to count on Iraq for stability in the oil market that might be undermined in the event of turmoil in SA (which is the main reason we've been so friendly to the Saudis even as they've promoted extreme Islamic fundamentalism). Finally, in the event the new Iraq decides not to stay in OPEC, that could bring the cartel down which would weaken autocratic regimes throughout the Middle East (SA in particular) and provide major impetus for reforms. (Although this last point may be a bit too optimistic.)
Of course the real linchpin (except perhaps in Iraq and Iran) is going to be progress in the Israeli-Palestinian conflict. It's hard to say whether (and how, exactly) a liberated and democratic Iraq will have much positive effect here. On the one hand, the government Sharon has put together does not look particularly good from the standpoint of progress with the PA. (Although on a different topic it's awesome to see the wacko Ultra-Orthodox parties out of the government.) The only hope was that he would have trouble forming a government before the war in Iraq, and that the war might somehow cause a Labor-Likud national unity government even though Labor had disavowed it before the elections. No such luck.
On the other hand, there is the chance that a strong display of benevolent Western force and subsequent democracy in Iraq could promote reform amongst the Palestinian leadership. Leading up to the first Gulf War, the main parties in the region to publicly root for Saddam were Jordan and the PLO. Chastened and moderated in the aftermath, three years later both were making peace with Israel (ahead of all the Arab states that supported the war).
Perhaps I'm being naive, but I'm not convinced that Sharon isn't willing to make peace with the Palestinians. His public position (and Bush's too, although less forcefully) is just that he won't make peace as long as Arafat is in power and the Intifada continues. While I don't agree with this position, it's quite justifiable considering Arafat launched the Intifada (as a bargaining chip and just when both sides were very close to a deal) and is clearly implicated in funding, arming and controlling the Al Aqsa Martyrs' Brigade which has used suicide bombers not just against soldiers and settlers but against civilians inside Israel proper. Point is, it's not unreasonable that a good result in Iraq could help leadership change (and moderation) in the PA, and in fact such a reform looks increasingly likely even disregarding Iraq. At that point, the ball would be in Sharon's court, and I don't know whether he'll do what he's said and pursue peace, or whether he's beholden to the settlers after all. One good point at least is that absolutely everyone knows exactly what a final deal will look like, since both sides came ridiculously close to one two years ago.
Combine the successful creation of a free, democratic, independent Palestine with a strong American pro-democracy push, and you've got a ton of impetus for democratic reform in Egypt, Saudi Arabia, Jordan and elsewhere, as well as at least a peace deal between Israel and Syria/Lebanon, if not a Syrian withdrawal from Lebanon (in which case that's another candidate for a very quick transition to democracy, considering it is sort of there already despite still being partially controlled by Syria) and even reform in Syria.
So that's the case. The Arab people desperately want democratization and economic liberalism; there's really little doubt about that. The problem is that they've been brainwashed into hating the United States. If the West (and the US in particular) does a good job in Iraq, then not only will that demonization begin to ring hollow, but the regimes responsible for it will lose a lot of leverage as well. Whether it will be enough for a cascade of democratic reform is impossible to say.
On the other hand, if the post-war Iraq goes badly or is seen in the Arab world as going badly; if the US acts badly (and even though I gingerly support him on this I wouldn't put it past Bush to do so) or is seen as acting badly...well, then all bets are off.