On the The number e page, one can find the following:
As far as we know the first time the number e appears in its own right
is in 1690. In that year Leibniz wrote a letter to Huygens and in this
he used the notation b for what we now call e. [...] Retrospectively,
the early developments on the logarithm became part of an
understanding of the number e.
Later:
So much of our mathematical notation is due to Euler that it will come
as no surprise to find that the notation e for this number is due to
him. The claim which has sometimes been made, however, that Euler used
the letter e because it was the first letter of his name is
ridiculous. It is probably not even the case that the e comes from
"exponential", but it may have just be the next vowel after "a" and
Euler was already using the notation "a" in his work. Whatever the
reason, the notation e made its first appearance in a letter Euler
wrote to Goldbach in 1731
Some other details are given here:
Euler started to use the letter e for the constant in 1727 or 1728, in
an unpublished paper on explosive forces in cannons, and the first
appearance of e in a publication was Euler's Mechanica (1736). While
in the subsequent years some researchers used the letter c, e was more
common and eventually became the standard.