That's actually what I was thinking. In older Japanese eras, people of lower classes didn't have last/family names (myo) unless they somehow earned them or were granted them as a favor by a daimyo. In that case, their name was turned into a surname and then passed onto their descendants. Perhaps, since the 3rd came from the early era of the Leaf, he didn't come from a noble family and distinguished himself by eventually becoming Hokage. From then on, he wanted to pass his name on as the family name to his children.

Granted, this is only speculation. But perhaps Asuma isn't directly related...he could always be a nephew or cousin, hence why he's not particularly close to Konohamaru.