The instrument clearly has 3 (三) strings (線), but where does 味 exactly fit in?
2 Answers
The instrument originated in China as the [三弦]{sānxián}, and it came to Japan via Okinawa. The Okinawan instrument's soundbox is covered with a snakeskin. The older Japanese name for it was 蛇皮【じゃび】線【せん】, literally "snakeskin strings". This instrument was introduced to the Osaka area from Okinawa during the 永禄【えいろく】 era (1558-1570). Over time, the jabi portion on the front shifted pronunciation to shami: the "j" became unvoiced "sh", and the "b" lost its plosive-ness to become nasal "m".
The 三味 spelling for the shami portion is an example of ateji, with some meaning overtones of jukujikun in the use of 三 to refer to the three strings, and (as far as I can tell) pure phonetics for the 味.
See also the Wiktionary entry (full disclosure: I edited that), and the monlingual Japanese sources at Kotobank and Weblio.
I'm not sure about this one as i am not a specialist in the question. But in my opinion it may be some kind of a meaning extension.
As 味 also means "a flavour", it might be that 三味線 is "three [good] flavour strings". And the meaning of a "[good] flavour" is just somehow extended to sounds made by the instrument. So it transforms in something like "three [good] sounding strings".
Not sure,though.