The Chief and the New Justice
First, without a second post to fill, the President may feel obligated now to nominate a strong conservative, like Michael Luttig, to the vacated post in order to appeal to the more conservative wing of the party and to keep the implied election promises that he made. If the Chief were stepping down, it could be that Bush could appease the right and moderate wings at the same time by appointing someone like Al Gonzales to be the Chief, and Luttig to the other vacancy. With The Post reporting that Gonzales' chances are fading, it appears that the Chief staying may have narrowed the choices which were available and made those choices more conservative. The Chief staying essentially leaves a 4-4 court, and the President might not need see as great a need to rush to have a full court seated. While 4-4 with a weak-kneed Kennedy is not ideal, it is more preferable than a 4-3 liberal split (If the CJ and SOC had left) or even a 4-4 split with the 4 conservatives including both Kennedy and O'Connor.
On the other hand, one could argue that the Chief staying should force Bush to nominate a more moderate judge, such as Gonzales or Edith Clement. This would possibly (though with Chuck Schumer manning the guns, probably not) avoid a bruising struggle and ensure that a full court was hearing cases come October.
I think in the end, the choice by the CJ to remain may not substantively impact the President's decision at all. The first reason is that Bush may have already been set on nominating a woman to replace O'Connor (starting a "female seat", much like the Brandeis seat once was for Jewish justices like Frankfurter and Goldberg - though I think technically Douglas took Brandeis' seat, it was in 1939, the same year Frankfurter came on the Court). The second reason is that the President may have been set on nominating a strong conservative anyway, but one without the "rough" edges that some may think Luttig has. Here, I would place two names, John Roberts first and Michael McConnell next. So in the end, it is quite possible that the Chief's decision has had no real substantive impact, but has only served to hasten the timetable for an announcement. My ultimate prediction is for the President to choose either Edith Jones or John Roberts.