Lieberman may have also been intended to appeal to Cuban Exiles and Florida Jews.
Lieberman may have helped with Cuban exiles- there were plenty of articles from the fall of 2000 about Lieberman's popularity with that group- and Jews in Florida, but if Gore really wanted to win the state of Florida outright, maybe there was a popular sitting Senator from Florida who had previously served as their Governor that could have helped to make a Gore candidacy much stronger statewide. Much as 2016 (at least to me), it seemed like Gore thought he could win with anybody as his running mate.