Remini's biography of Daniel Webster says that with a few tweaks Webster would have been Harrison's Vice-President instead of Tyler.
Assume Harrison still dies weeks after his inauguration. What next?
OTL the election of 1840 was a a major Whig victory but Tyler frustrated most of their agenda. Webster will be onboard with the "American System." He is in favor of internal improvements, protective tariffs, the Bank of the United States (which failed to be renewed under Tyler), Western land settlement at very cheap rates, and is opposed to Texas annexation and, mildly, to the expansion of slavery (he is opposed to Texas annexation because he thinks it will enflame the slave issue and cause sectional division). He is a statesman and extremely willing to compromise in the interests of creating a long-term settlement that will not be reversed by the next political swing. He is also fairly corrupt. He is a magnificent orator whose speechs are even more impressive in writing and therefore in newspapers.
So what next? Does he create some sort of compromise on the Bank that leaves it viable (OTL he proposed renewing the bank but allowing individual states the authority to keep it out of their demesne if they wished)? Does he bow to the strong national pressure for Texas annexation, but in a way that avoids war with Mexico? Does he resist Texas annexation, and does this just lead to later annexation or to long-term Texas independence?* Does he settle the Oregon question early?
*If it does, this might create the precedent for independent American republics in the southwest, like a Bear flag Republic and a Deseret.