One interesting idea was an Eisenhower-Taft ticket. Senator Alexander Smith of New Jersey had been pressing Taft to take the job if Ike offered it, and according to Smith's diary, Taft had been "willing to consider the matter if it would help to heal the wounds and enable the party to move forward." When the Eisenhower camp--Herbert Brownell, Sherman Adams, Henry Cabot Lodge, Senator Frank Carlson of Kansas, and others--met to consider a running mate, Smith joined the group and immediately suggested Taft. Adams later wrote that Brownell and Lodge "made no move to oppose it", and Arthur Summerfield, incoming GOP national chairman, "thought it was a fine idea."
https://archive.org/stream/firsthandreport017252mbp#page/n57/mode/2up
Ike, however, had other ideas: he had already told associates that he favored a younger man like Knowland, Governor Alfred E. Driscoll of New Jersey, or especially Nixon. In any event, at the meeting, Sinclair Weeks (later to be Ike's Secretary of Commerce) argued that Taft would be more useful in the Senate. And Russell Sprague, a veteran Dewey aide, argued that with Taft on the ticket, Ike would lose New York (which in retrospect is clearly not true, given that Ike was to carry New York by almost 850,000 votes--but of course nobody could know that in advance). With this remark the group turned to other names, and quickly settled on Nixon. (My source for this is James A. Patterson, *Mr. Republican: A Biography of Robert A. Taft*, pp. 564-5.)
Of course if he had chosen Taft, and the latter had died of cancer on schedule, the country would be without a vice-president for most of Ike's first term, and after the 1954 election, the nation would be one heartbeat away from having a Democrat (Sam Rayburn) as president.
Anyway, in OTL in 1956, some Republicans (e.g., Harold Stassen) thought that in view of Eisenhower's health, the vice-presidency would be a major issue of the campaign, and suggested that the controversial Nixon would hurt the party. Stassen wanted Ike to dump Nixon in favor of Governor Christian Herter of Massachusetts.[1]
https://en.wikipedia.org/wiki/Christian_Herter Eisenhower himself offered Nixon a cabinet position (supposedly so that he could "gain executive experience" as a preparation for a 1960 presidential run) but Nixon turned him down, realizing that this would be seen as a demotion and harm his future prospects. If Nixon had agreed, Ike was considering a conservative Democrat or ex-Democrat as running mate--most likely former Secretary of the Navy and future Secretary of the Treasury Robert B. Anderson, whom Ike greatly admired. Another possibility was the conservative Democratic senator from Ohio, Frank Lausche
https://en.wikipedia.org/wiki/Frank_Lausche who might attract Catholic voters.
[1] Herter was born in France of American expatriate parents but I doubt that this would be a major problem. The opinion of most lawyers was that Americans born of American parents living abroad were "natural born citizens" of the US within the meaning of the Constitution. Perhaps a more serious obstacle was Herter's health: "he suffered from severe arthritis for the last twenty years of his life [Herter died in 1966] and frequently had to use crutches." Michael S. Mayer, *The Eisenhower Years*, p. 299 With Ike's health an issue, a younger and more vigorous running mate might be preferable.