Does anyone have any ideas on Clay's cabinet?
Clay would obviously much rather be his own Secretary of State, so he'll probably choose a loyal nonentity - maybe William Segar Archer, chair of the Senate Foreign Relations Committee?
One obvious choice for Clay's Secretary of State would be the man who would eventually get that job under Taylor--John Clayton.
Clay had in fact very much wanted Harrison to put Clayton in the Cabinet:
"As the party's most prominent congressional leader, Clay expected an offer of the State Department in Harrison's cabinet, but he declined the post even before it was tendered because he preferred to remain in the Senate. With his blessing, Webster received that important slot instead. But Clay hoped to staff the remainder of the cabinet with his friends. He succeeded to the extent that he blocked the rumored appointment of Charles Wickliffe, a Kentucky rival, and secured instead the attorney generalship for his faithful lieutenant Crittenden. John Bell of Tennessee, the secretary of war, and Thomas Ewing, the secretary of the Treasury, could also be considered Clay allies.
"Clay was especially anxious for the appointment of John M. Clayton of Delaware. Initially, Clay had suggested Clayton for the Treasury and Ewing for the postmaster general's office. But Webster had prevailed on Harrison to appoint New York's Francis Granger as postmaster general instead and to switch Ewing to the Treasury Department. Frantic to get Clayton into the cabinet, Clay then held a stormy personal interview with Harrison, demanding Clayton's appointment as navy secretary, the remaining cabinet post. Proud and vain, Harrison was determined not to appear subservient to the party's senior statesman and reportedly reminded Clay that he, rather than Clay, had been elected president. Instead, Harrison chose to let Whig congressmen from the South Atlantic states fill the last spot, and they decided on George Badger of North Carolina, who was also friendly to Clay. Thus, despite Clay's failure to win a place for Clayton, his heated argument with Harrison, and Webster's appointment to the most important post, Clay had done well. The other cabinet members were either his outright supporters or at least personally cordial to him..." Michael F. Holt,
The Rise and Fall of the American Whig Party: Jacksonian Politics and the Onset of the Civil War, pp. 124-5.
https://books.google.com/books?id=hMkYklGTY1MC&pg=PA124
I have a feeling that a number of members of the Harrison Cabinet (which of course, except for Webster, resigned en masse after the Tyler-Clay split became unbridgeable) will reappear in the Clay Cabinet. (But maybe he will want Crittenden to stay in the Senate to lead the Whigs there?)