Could I propose James F. Byrnes, to be Vice President nominee for Joseph Kennedy.
Uh, no. Byrnes was a vocal opponent of *Brown v. Board of Education.* There is no way he can get the Democratic nomination for vice-president. Worries about his effect on labor, liberal, and African American voters led FDR to reject him as running mate even in 1944, when Byrnes was much more a mainstream Democratic figure than he would be in 1960. Oh, and Byrnes would be 78 years old by 1960.
Even LBJ as running mate was somewhat controversial, even though he had not signed the Southern Manifesto (against *Brown*), had gotten the first twentieth century Civil Rights law through the Senate in 1957, etc. LBJ was as far to the right as a plausible Democratic vice-presidential candidate could be in 1960.
Oh, and another thing about Byrnes: he was born a Catholic and later became a Protestant. This was one of the things that led FDR to reject him in 1944--he thought Byrnes might alienate both Protestants and Catholics. A ticket of one Catholic and one ex-Catholic was inconceivable in 1960.
JPK could only get the Democratic nomination if he repudiated his youthful isolationism and had moderately liberal politics more or less similar to those JFK had in OTL by 1960. This would mean he would have pretty much the same range of choices of running mate: LBJ, Scoop Jackson, Symington, maybe a Midwestern governor, maybe a southern moderate other than LBJ (like Albert Gore, Sr., who had also refused to sign the Southern Manifesto and had also voted for the 1957 Civil Rights bill).