The "I" does not refer to any specific in-show character. The show takes its name from the opening line of the Stephen Foster song Jeanie with the Light Brown Hair, from 1854:
The opening line notably got used as the basis for the title of the TV
series I Dream of Jeannie.
Here it is on Youtube.
The opening line is "I dream of Jeanie with the light ...
However, I am wondering: was the show already suffering low ratings and the marriage plotline a desperate attempt to revive it, or were the ratings doing fine before the marriage?
It should be understood that I Dream of Jeannie wasn't a massive ratings hit at any point in its five-year TV history.
Wikipedia indicates that for 3 of its seasons it wasn't in ...
There was a promotional contest for fans and viewers that determined the final combination to the safe.
From I Dream of Jeannie (emphasis mine)
In a third-season four-part episode ("Genie, Genie, Who's Got the Genie?" January 16 – February 6, 1968), Jeannie is locked in a safe bound for the moon. Any attempt to force the safe or use the wrong combination ...