string, cord, samisen music
絃 centers on a thin, flexible line used for tying or producing sound: a string or cord. The samisen music sense is a natural extension, referring to the instrument's strings.
絃 combines 糸 (thread) with 玄, which likely contributes the sound and may relate to dark or mysterious, but the exact historical development is uncertain.
The left side 糸 is thread, and the right side 玄 looks like a twisted strand. Picture a thread being twisted into a strong cord used for a samisen's strings.
For ゲン, imagine a genie emerging from a twisted cord: genie -> ゲン, and the cord vibrates with magical music.