First a question, does R. Abaye mean it was known for having many murderers at the time it was given to Menashe, or in his own time? Note that it does not say they "built" the cities, but "rebuilt them." If he referred to the time of the conquest, then the tribes were supposed to conquer or drive out any men that remained in the land.
But it was not just one town we are talking about. The text makes it clear that the land itself was a major reason why Gad and Reuven wanted territory in the region, so presumably this is one reason why it was attractive to Menashe as well.
Now the Reubenites and Gadites had a very large number of livestock.
Noticing that the land of Jazer and of Gilead was a place suited to
livestock, 2 the Gadites and Reubenites came to Moses and Eleazar the
priest and to the leaders of the community and said, 3“The region of
Ataroth, Dibon, Jazer, Nimrah, Heshbon, Elealeh, Sebam, Nebo and
Baal-meon— 4 the land which the Lord has laid low before the community
of Israel, is a land for livestock, and your servants have livestock.”
Also, Menashe was not only given one particular town, but a major part of the region of Gilead. In fact, we know of two towns with the name of Gilead included in their designation (Jabesh-gilead near the Jordan and Ramoth-gilead further east). The map below makes clear why Menashe wanted it: it was a very large region, virtually equaling the size of the large territory they were allotted west of the Jordan. It made them the largest tribe in terms of land. So it was a major bonus, including not only the two towns mentioned above, but several others, as well as rivers, pasturage, trade routes and other natural resources.
So even if ancient Jabesh-gilead was known to R. Abaye for having many murderers, that problem was more than made up for by the territory and other towns given to the tribe. In addition, we may consider that other towns - especially those in Canaan proper - had other problems just as thorny and perhaps more so, such as idolatry and various other forms of immoral behavior.