I don't know if this counts by your, but Grover Cleveland had a different VP during each of his terms. Andrew Jackson also had two different VP's, but Van Buren wasn't his VP until March of 1833. Another was FDR, who had three different VPs during his 12 years in office. Garner was his VP for the first two terms, then Wallace, and finally Truman in 1945. Im not sure if there are any others, but here are the most notable three.

Technically John Adams did not run with his Vice President Thomas Jefferson as his running mate, but he hadn't picked him as his running mate the first time, either - Jefferson ended up finishing second because of a plot by Alexander Hamilton to make Thomas Pinckney President while Adams would be Vice President again was foiled by electors trying to prevent it by casting second electoral votes for someone other than Pinckney, causing Jefferson to finish second instead of Pinckney. Adams ran with Charles Pinckney in 1800, and was beaten by Jefferson and Aaron Burr, who tied.

Thomas Jefferson dropped Aaron Burr after his first term (replacing him with George Clinton), after Burr had attempted to manipulate the House of Representatives to break the 1800 electoral college tie in his (Burr's) favor, even though he had been nominated by the party to be Jefferson's Vice President (this led to the 12th Amendment, which changed the electoral college from casting two votes each for President, to casting one vote for President, and one for Vice President). Jefferson won reelection in a landslide. Burr, who had already murdered Alexander Hamilton earlier in the year, went west and maybe tried to become Emperor of Mexico.

The next Vice President to be replaced on the ballot was Lincoln's VP, Hannibal Hamlin, who the Republicans (running in 1864 as the "National Union" party) replaced with Democrat Andrew Johnson. Lincoln, of course, won reelection and was assassinated a month after his inauguration, resulting the the Democrat becoming President.

Franklin Roosevelt was the next to replace his Vice President, doing it twice - first replacing John Nance Garner with Henry Wallace when he ran for his third term in 1940, then replacing Wallace with Harry Truman in 1944. FDR obviously crushed all before him.

Other Presidents ran for re-election with a Vice Presidential nominee other than their Vice President from the first term, but in these circumstances, their previous Vice President had either resigned or died in office during their first term, and there was no manner to replace a vacancy in the Vice Presidency until the 25th Amendment was ratified in 1967.