This article is from the alt.usage.english FAQ, by Mark Israel firstname.lastname@example.org with numerous contributions by others.
The OED erroneously states that Julius Caesar was born by
Caesarean section. Merriam-Webster Editorial Department (on its AOL
message board, in response to a query from me) writes:
"The name 'Caesar' is a cognomen, a nickname given to one member
of a Roman clan and borne by his descendants as a kind of surname.
No one knows who the original Caesar was, but his descendants
within his clan, the Julii, continued to use his cognomen and formed
a major branch of the clan.
"According to a legend related by the Roman naturalist Pliny,
the first Caesar was so called because he was cut from the womb of
his mother ("a caeso matris utero"), "Caesar" supposedly being a
derivative of the verb "caedere" 'to cut'. This etymology is
dubious, but the name 'Caesar' has continued to be associated with
surgery to remove a child that cannot be delivered naturally.
"The OED gives evidence for the belief that Julius Caesar, the
most famous bearer of the cognomen, was delivered this way that
dates from 1540. There is no authority for this notion in ancient
sources. Moreover, Julius Caesar's mother lived long after his
birth -- unlikely if she had undergone such an operation, which few
women would have survived in those days. In any case, the earliest
record we have for the term 'cesarean section' used in English dates
from 1615. You can easily see from these dates why we say that the
term came from the belief, and not, to throw in a little more Latin,
The Facts on File Encyclopedia of Word and Phrase Origins
suggests that Caesar's name may have become associated with the
operation because of an edict of the Caesars of Imperial Rome (Lex
Caesarea) that any pregnant woman dying at or near term was to be
delivered by C-section; but Merriam-Webster Editorial Department
says "We can find no evidence for" such an edict.
Also not named directly after Julius Caesar are "Caesar salad"
(allegedly named after a restaurant named Caesar's in Tijuana,
Mexico); and "Julian day" (number of days elapsed since 1 January
4713 B.C., used in astronomy; named by Joseph Scaliger after his
father, Julius Caesar Scaliger). The computer term "Julian date"
(date represented as number of days elapsed from the beginning of a
chosen year) was apparently inspired by "Julian day".