The American Heritage Dictionary gives the following etymology:
Obsolete Dutch monssoen, from Portuguese monção, from Arabic mawsim, season, from wasama, to mark
That seems like a reasonable etymology. But from there, it links to the Semitic root "wsm" and writes:
To be(come) fitting, suitable. 1a. mazuma, from Medieval Hebrew mezumman, fixed currency, from Mishnaic Hebrew mezumman, fixed, passive participle of zimmen, to arrange, arrange a meal, invite, denominative from Hebrew zeman, appointed time, season, from Aramaic zeman, time; b. Sivan, from Hebrew sîwan, a month name. Both a and b from Akkadian simanu, season, time, name of a month corresponding to parts of May and June, from (w)asamu, to be(come), fitting, suitable. 2. monsoon, from Arabic mawsim, season, from wasama, to mark, wasuma, to be(come)beautiful
This is a bit of a balagan, to say the least! While mezuman and Sivan are connected to the Hebrew root זמן, what does wasamu have to do with it? Unless I can find a better explanation, I'll assume that the editors of the dictionary made a mistake.
Does wasama have a Hebrew cognate? It seems to be connected to the word for "name", שם shem. As Robert R. Ratcliffe writes here:
The CA verb yasimu/wasama "mark, distinguish" was possibly derived at an earlier period from a noun meaning "name" (CA /ism/) which can be reconstructed as a two-consonant stem for both Proto-Semitic and Proto-Afroasiatic.
After listing cognates for shem in other Semitic languages, Klein writes the following:
Some scholars connect the above names with Arab. wasama (= he branded cattle,
stamped, marked, branded), wasm (= branding cattle, stamp, mark, brand).
We see a sign of this root in the Aramaic and Hebrew word for "wart, mole, mark" = shuma שומא/ שומה (see Niddah 46a, Ketubot 75a). Jastrow also connects this meaning of shuma to the sense of שום, meaning "to value, estimate" and the source of the word shamai שמאי - assessor. Klein, however, says that sense of שום comes from a different Semitic root, meaning "to buy".