The story
From Georgian ენგური, derived from Old Georgian ეგური (eguri), the exact origin of which is disputed. According to one theory, the hydronym is from Mingrelian გური (guri, “heart”), however, other sources derive it from Proto-Georgian-Zan *egr- (> *egur-), whence also ეგრისი (egrisi, “Egrisi”), მეგრელი (megreli, “Mingrelian”). It has also been connected with Mingrelian გერი (geri, “wolf”).
Etymology from Wiktionary, CC BY-SA 4.0.