To whatever extent it lasts, Rudy Guiliani's presidential campaign could challenge some aspects of the presidential nomination process. For one thing, I don't know of any mayor that has ever made a serious run for president. The precedent this sets could do any number of things. For instance, it could open up a whole new field for recruiting candidates. Being mayor may not give the experience that serving in the congress does, but you could argue that being mayor of a large city (especially New York) is a better training ground for the presidency than being governor of some states (especially Texas). It could establish for the time being that we don't think having been mayor is good enough experience to be president.
The other thing that Guiliani could really affect is how republicans choose their candidate. I've written before about how the party's right wing has disproportionate influence over its candidate. So far Guiliani is doing surprisingly well in spite of that. If he gets the nomination, if he even lasts very long, it might well be seen as a rebuke of the social conservatives who foist their politics on everyone else who wants to vote GOP.
What's going to put him in direct conflict with the social conservatives is his support for abortion rights, his (as far as I know) reasonable view on civil rights for gay and lesbian people, and his colorful personal life (multiple marriages, an ugly divorce and well-known mistress during his mayoral term, etc). In terms of the abortion rights, though, he's not only set himself apart from the other republican candidates simply by supporting them, he's taken a stand beyond that of any democrat. According to CNN.com, he reiterated that he supports the idea of having public money pay for them for poor people.
This is a surprisingly pragmatic view of abortion, one that could either sink his campaign or draw an apology from someone with less backbone. He's basically saying that abortion, even though he takes the predictable I'm-personally-opposed-to-it line, is a medical procedure (one of the most common ones, in fact), and as such it ought to be a part of any health care plan for low-income people. The Hyde Amendment banned federal funds from paying for abortions (through Medicaid) as of 1977. Since then, all but 16 of the states have passed similar bans involving state money. Texas is one of them, although there is a city program that covers them in Austin.
At any rate, that's a bold stand he takes. Incidentally, it will also be interesting if at any point the fact that, prior to 9/11/2001, he was an unpopular, divisive mayor becomes an issue.