I've seen quite a few of the names I. Bernard Cohen in the history of science book. But I couldn't find what I. meant.
I am quoting the full footnote published in Seven Decades of History of Science: I. Bernard Cohen (1914–2003). The full text is available in JSTOR Link.
Cohen’s sister, Harriet, was three years older than he; she died in 1994. The “I” in Bernard Cohen’s name has always been a mystery—one he insisted was no mystery at all, remarking to friends that it was like the “S” in Harry S. Truman. Replying to one author, Cohen complained that he had “invented a fictitious new name for me, putting it within square brackets as if I had chosen to hide it and you were revealing it to the public. What the basis for your invention could possibly have been, I cannot imagine. Certainly, recourse to any of the standard sources of bibliographic information would have prevented you from making this error. Had you consulted the Library of Congress catalogue, the National Union Catalogue, or Who’s Who, or any other proper source, you would have found that my correct and full legal name is (and always has been) and is listed as, I. Bernard Cohen, which is the name recorded on my birth certificate, my passport, my FBI clearance, every copyright for some thirty or more books, and other official records”: I. Bernard Cohen, letter dated 9 Mar. 1992 (courtesy of Susan T. Johnson).
Another online Encyclopedia claims I is for Ierome without any reference. Most likely this is a conjecture. All his publications just use I. so apparently I. has no meaning. It is just a sound.
It's impossible to know for sure, but I. Bernard Cohen's parents were Isidore Cohen and Blanche Bernstein (1920 Census). Isidore's parents were Israel Cohen and Harriet Hyam (Isidore's marriage record). I. Bernard's sister Harriet was named (as traditional) for her father's mother. It stands to reason that I. Bernard would be named for his father's father: Israel.
However, as he claimed, there seem to be no written records remaining to support this idea. He was known as Bernard from a very young age.