OED:

1905, "of an indefinite number," originally Morse code slang for "dash," influenced by association with numerals such as twenty, thirty, etc.; umpteen (1917) is World War I army slang, from umpty + teen

However, the disarmingly entertaining Uncyclopedia (qv) claimeth:

Umpteen is so called because it is ten plus ump. Ump was defined as umpteen minus ten by the Pythagoreans.

(

*ump*tangentially being defined by the same source thus: Ump (or "umpt") is the base unit of the third axis of a three-dimentional number graph. Just as "math" (the base unit of the second axis on a two-dimentional number graph) is the square-root of math, ump is the square-root of umpteen)

At last my tired tagline gets the ball