I presume the name stands for "五人碁" - "five men" (Chinese) "Go" (Japanese). Now, it quickly has evolved away from the number five. As it happens, there is a perfect Chinese equivalent to that evolution, the character 伍, also pronounced "wu" in Chinese, which originally means "five men squad", but has come to be more generally uses, as in "ranks" or "company". This invites replacing 五 with 伍, which would then leave the "ren" redundant. Since the word is already a mix between Chinese and Japanese, the "ren" could as well be understood as Japanese 連, forming "rengo" (連合) together with the third syllable. The whole name thus could represent "伍連合" in Chinese-Japanese mix.

BTW, the character 伍 is also used in Japanese, where it is pronounced "go", so that an entirely Japanese pronunciation would be "gorengo".