In November of 1861 virtually no one in high command has any real experience, one way or the other. They also have far fewer war-time political connections to call upon (this does not count the peace-time political connections many people used to get appointed Generals in the first place).
Therefore, whosoever replaces McClellan - as AotP commander, not as anything higher as McClellan was for a while - will be based mostly on seniority.
A possibility is to promote one of the current senior commanders from the AotP. The highest is Irwin McDowell, but he had his chance at Bull Run, so a promotion back to army command is unlikely. More probable is "Bull" Sumner, with runners-up Samuel Heintzleman [sic?] and Erasmus Keyes. I forget if McClellan had organized the AotP into corps yet, in which case these three would be outright the senior-most generals. If he had not, then bearing in mind that Heintzleman & Keyes were truly political appointees, McClellan himself had preferred Fitz John Porter and William Franklin. Depending on the exact nature of McClellan's death, his preferences may or may not carry any weight. Regardless, within the AotP itself, these 5 (with the first 3 more likely) are the viable possibilities.
None of these 5 seem like inspired choices, with Sumner, Porter, and Franklin slightly preferable. Regardless, they are competent enough to not suffer any huge disaster, and cautious enough not to gain any major victory. Admitting the butterfly net will be humongous from this early a PoD, I would not predict a drastically shorter or longer war.
Another option is to bring in an outsider to take command. Seeing as the AotP is the largest, and seen as the most important, army the Union has, whoever comes in likely must be of an army- or department-commanding rank already. Thus, for instance, while I would love to see Thomas, as mentioned by Jonathan, there is no way that will happen. He was a local divisional commander under department head Don Carlos Buell, in turn serving under regional commander Henry Halleck.
As such, Halleck himself is a reasonable first choice, bringing him in from out west. If he does not come, then either Buell (commanding in eastern Kentucky) or Grant (commanding in western Kentucky) are possibilities.
As for other departments....one option is William Rosecrans, commanding in West Virginia. As he had already served under McClellan in the earlier campaigning there, and performed very well, he is a very deserving black-horse candidate.
Another option is John Fremont, commanding in Missouri. As a darling of the Radical Republicans, his appointment would be very polarizing in terms of whether or not it could happen at this early stage in the war.
Coming in last on the list of possibilities are choices such as Ambrose Burnside, depending on how high a command he officially had already and how far preparations for the Roanoke Expedition are coming along. In a similar situation is Benjamin Butler, depending on preparations for New Orleans.
Of these outsiders, the top tier obviously consists of Grant, with Rosecrans a close (IMO) second. Probably none of the others would be outright disasters - regardless, I could see this shortening the war substantially, depending on how the butterflies happen.
A point to consider for these outsiders, however, is who takes their place. If Grant, for instance, comes east, then someone else is in charge of the Forts Henry-Donelson expedition, and this person will likely be less successful than Grant. Even if they take the forts, they will not likely capture the Confederate defenders, which can cause huge changes later down the line, starting with the extra 10,000+ troops leading to a decisive Confederate victory at Alt.-Shiloh.
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Also, to close down the speculation before it starts, the following people will *definitely not* take McClellan's place (no matter how much we may like some of them), as they are far too junior at this point in time:
George Thomas, George Meade, William Tecumseh Sherman, Joseph Hooker, John Reynolds, Darius Couch, Winfield Scott Hancock, Dan Sickles, Gouvernor Warren, John Sedgewick, Phillip Kearny, William "Baldy" Smith, Oliver Otis Howard, Slocum, Ord, Sheridan, McPherson, Wright........
[While I think that Franz Sigel, John Pope, and Nathaniel Banks also belong on the above list, I could be persuaded to move them to the "Outside shot" category.]