He was also in poor health by 1952.
An interesting archived article on the offer:
http://www.time.com/time/magazine/article/0,9171,817454,00.html
However, while he may not have been an active Zionist, he didn have some empathy with the young state of Israel. Certainly he and Chaim Weizmann, Israel's first president, were close personal friends, and he bequeathed his papers to Hebrew University - on whose board of governors he served.
http://en.wikipedia.org/wiki/Albert_Einstein's_political_views#Zionism
Perhaps if Einstein was in better health and Weizmann made the request personally to take the role on after his death?
Another POD could revolve around the founding of Brandeis University. Einstein was heavily involved with the early stages and the school was initially to be called Einstein University. However, Einstein clashed with other founders and agreed with many choices, including the selection of Abraham Sachar as the first dean.
http://en.wikipedia.org/wiki/Brandeis_University#Einstein_incident
Were Brandeis University to become Einstein University and his public involvement with the school in its first few years thorough, there are a number of interesting effects. The school - which also attracted Eleanor Roosevelt and Leonard Bernstein to its staff in its first decade - would be even more prestigious earlier, but also perhaps even more closely aligned with the academic Left.
However, perhaps Einstein's closer involvement with institutional Jewishness (and early Brandeis was very much institutionally Jewish) could have led him into a frame of mind to take the offer to be Israeli president.
Effects? This may ironically nip the Israeli nuclear program in the bud, given Einstein's views. It may also raise Israel's political prestige in the community of nations. Finally it may keep the USSR courting Israel longer: I'm not certain they had firmly picked a horse in the Middle East wars until Nasser came to power in 1954.