This period of time came about when I was 8 years old, so I don't recall a whole lot about politics at that point. That said, one likely cabinet member would have been Maryland governor Theodore McKeldin: he was respected and rather well thought of (I believe he might have made the short list of running mates for Nixon in '60, in fact), so his nomination as, perhaps, attorney general doesn't seem too far-fetched to me.
Going a bit further, it doesn't seem out of the realm of possibility that Ike's last Secretary of State, Christian Herter, would have been kept on. Perhaps long-time California Senator William Knowland might have been appointed to head the Department of the Interior.