The word, Zanj, to the arabs actually translates{roughly} to the "country of the blacks". It was a term used by medieval Arab/Muslim geographers and sailors alike to refer to certain parts of what is today known as East Africa{mainly the Swahili coast and adjacent lands}.
The word Zanj was also deployed when referring to the bantu inhabitants of this region.
So, to go straight to your question, Bahr-el Zanj would roughly translate to the "Sea of the land of the blacks"(mouthful) or "Sea of the blacks". That is if translations are to be strictly adhered to.
Note: Bahr-el Zanj is a former name for the western portion of the Indian Ocean adjacent to the shores of south east Africa. It wasn't a general designation for the Indian Ocean. For certain reasons, the Arab sailors were really scared of the western portion of this ocean hence the need to mark it out specifically.
The terms Bahr-el Zanj and Zanzibar both share similar origins.
The word Zanzibar came from Arabic zanjib?r [zand?iba?r]), which is in turn from Persian zangbâr [zæ?ba:r]), a compound of Zang [zæ? ], "black") + bâr [b?:r], "coast"),[14][15][16] cf. the Sea of Zanj. The name is one of several toponyms sharing similar etymologies, ultimately meaning "land of the blacks" or similar meanings, in reference to the dark skin of the inhabitants.
https://en.wikipedia.org/Zanzibar