I thought this was an interesting pshat from Erel Segal-Halevi (a fellow Yodeyan!).
In context, the psukim (Yeshayahu 54:11-17) are talking about building Jerusalem. So bonayikh means the builders of Jerusalem. Later on (Yeshayahu 60:10), the pasuk says
וּבָנוּ בְנֵי-נֵכָר חֹמֹתַיִךְ
And aliens shall build up thy walls
From this, we learn that non-Jews will be among the builders of Jerusalem.
So while we might have thought that our pasuk was saying that the children of Hashem would become learned in God's Torah and therefore they would have a great peace, instead, read the second בניך as בוניך, "your builders". That includes the non-Jews as we learned from perek 60. Learning Torah doesn't just cause שלום for the Jews; it causes שלום בעולם.
If we had read the second בניך as it's written, we would have thought the pasuk was only implying peace for the Jewish people and therefore it would not prove R' Chaninah's point.