Leonhard Euler (1707-1783) introduced e for this constant in a manuscript,
Meditatio in Experimenta explosione tormentorum nuper instituta (Meditation
on experiments made recently on the firing of cannon), written at the end
of 1727 or the beginning of 1728 (when Euler was just 21 years old). The
manuscript was first printed in 1862 in Euler's Opera postuma mathematica
et physica, Petropoli, edited by P. H. Fuss and N. Fuss (vol ii, pp.
800-804). The manuscript describes seven experiments performed between
August 21 and September 2, 1727:
For the number whose logarithm is unity, let e be written,
which is 2,7182817... [sic] whose logarithm according to
Vlacq is 0,4342944... [translated from Latin by Florian Cajori].
Euler next used e in a letter addressed to Goldbach on November 25, 1731,
writing that e "denotes that number whose hyperbolic logarithm is = 1."
The earliest appearance of e in a published work was in Euler's Mechanica
(1736), in which he laid the foundations of analytical mechanics (Maor,
p. 156).
Maor writes (page 156):
Why did he choose the letter e? There is no general consensus.
According to one view, Euler chose it because it is the first
letter of the word exponential. More likely, the choice came
to him naturally as the first "unused" letter of the alphabet,
since the letters a, b, c, and d frequently appear elsewhere
in mathematics. It seems unlikely that Euler chose the letter
because it is the initial of his own name, as occasionally been
suggested: he was an extremely modest man and often delayed
publication of his own work so that a colleague or student
of his would get due credit. In any event, his choice of the
symbol e, like so many other symbols of his, became universally
accepted.
Ball says: "It is probable that the choice of e for a particular base was
determined by its being the vowel consecutive to a."
In a post in sci.math in 1995, Wei-hwa Huang wrote: "I believe that e
was not named because it was the first letter in Euler's name, but rather
because he was using vowels for constants in a proof of his and e happened
to be the second one."
Several textbooks claim that the letter e was chosen to honor Euler.
Cajori has no information to support this claim. The early uses of symbols
for 2.718... mentioned by Cajori are as follows:
1690 b Leibniz Letter to Huygens
1691 b Leibniz Letter to Huygens
1703 a A reviewer Acta eruditorum
1727/8 e Euler Meditatio in Experimenta explosione tormentorum
nuper instituta
1736 e Euler Mechanica sive motus scientia analytice exposita
1747 c D'Alembert Histoire de l'Acadimie
1747 e Euler various articles
1751 e Euler various articles
1760 e D. Bernoulli Histoire de l'Acadimie r. d. sciences
1763 e J.A. Segner Cursus mathematici
1764 c D'Alembert Histoire de l'Acadimie
1764 e J.H.Lambert Histoire de l'Acadimie r. d. sciences et d.
belles lettres
1771 e Condorcet Histoire de l'Acadimie
1774 e Abbi Sauri Cours de mathimatiques
1775 e J.A. Fas Inleiding tot de Kennisse en het gebruyk der
Oneindig Kleinen
1782 e P. Frisi Operum tomus primus
1787 c D.Melandri Nova Acta Helvetica physico-mathematica
The term Napier's constant has been suggested for 2.718... The name
Euler's constant may be inappropriate for this number, as the number
was known before Euler's birth and Euler's constant more frequently
is used to refer to 0.577..., also called the Euler-Mascheroni constant.
</quote>
Reference: http://members.aol.com/jeff570/constants.html