An umpire is, in the oldest sense of the word, someone who is not equal to either side.
The word comes from Old French nonper, built from Latin non, "not," and par, "equal." Etymonline says it meant "odd number, not even," and was used for a third person who settled disputes between two parties. Merriam-Webster glosses the Anglo-French form as "single or odd," so the original sense was roughly "one without equal."
It appears in English in the mid-1300s as noumper, and its first use was legal: an arbitrator or mediator, someone who decides when others cannot agree.
Then a small slip changed it. In Middle English the word was preceded by the article "a," so people said a noumpere. Over time, listeners heard this as an oumpere, as if the n belonged to the article. Linguists call this misdivision or metanalysis. By around 1400, Etymonline notes, the n had gone. Late 14th-century copies of Piers Plowman show a range of spellings, including nounpere, ompere and nompeyr, which captures the word in the middle of the change.
The same slip happened in other words. Etymonline lists apron, adder and auger as words that lost an initial n through misdivision.
The meaning changed too. Games came next: Etymonline records the sense of an arbiter in a contest, first in wrestling by 1714, and sports officiating became the main use. The verb appears in the 1590s, meaning to appoint an umpire, and in the 1610s for deciding as one. Merriam-Webster dates its verb sense to 1609.
The two dictionaries differ a little on the noun. Etymonline says mid-14th century, Merriam-Webster gives the 15th. Neither disputes the French source or the lost n.
It suits the job. The word for the person who must favor neither team started out meaning "not a match" for either.