At the time (I'm old enough to remember Ike as president--barely), Ike favored Thomas Dewey. In 1956, he was 54 years old: pretty much par for the course. He had the credentials as a governor of New York. Trouble is, he also had the baggage of 1948 and 1944 (although the latter, IMO, shouldn't count since that was a wartime election and his opponent was Franklin Roosevelt). The memory of 1948 was still very strong with a lot of GOP bigwigs, so unfortunately Dewey was a non-starter.
Herter wound up succeeding Foster Dulles at State IOTL. Maybe he'd be acceptable as an Ike successor. Nixon...at age 43, the youngest elected president. If he'd gotten the backing of Dulles and a few of the more conservative members of the administration, he might just have pulled it off with perhaps Herter or possibly a young undersecretary named Nelson Rockefeller. I believe by then Harold Stassen was well on his way to becoming something of a joke, so I doubt he'd have been in the mix.
You can bet that Adlai Stevenson would have gotten the Dems' nod. His VP choice might be different, though: possibly John F. Kennedy might have gotten the nomination as opposed to Estes Kefauver. (There's an interesting prospect: two zillionaires running for the vice presidency.)