As exscape already mentioned, the symbol I was already taken for current (Ampère referred to electric current as "l'intensité du courant électrique", and used the symbol I in his equations).
According to a bio for Heinrich Lenz on the website of the National High Magnetic Field Laboratory (which is affiliated with the Los Alamos National Laboratory),
Lenz’s name, or at least his first initial, is attached to still one
more area of physics nomenclature. The symbol “L” was chosen to
represent “inductance” in honor of his pioneering work in