Then he [Abraham] said, "Oh let not the Lord be angry, and I will speak again but this once. Suppose ten are found there." He [God] answered, "For the sake of ten I will not destroy it."
Lot and his family had no significant impact for good upon Sodom and Gomorrah. Granted, Noah lived a life of righteousness and was unable to save his society. The evil of Noah's day was profound and unmoved by his preaching and Godly example. Therefore, one cannot necessarily state that it was Lot's fault that Sodom and Gomorrah had no respect for him.
The difference, though, between Noah and Lot is that Noah saved his family. Noah's family so respected him that they joined him in his adventure, though the world mocked. Building a ship in the desert and talking about rain when there had been no rain before was a lot for his family to swallow. Still, they followed.
In Lot's household there were the following members:
That is ten persons. There were possibly grandchildren. [Uncle Abraham was 99 at this time.] There were likely servants and probably many servants. Lot had entered Sodom as a very wealthy man. There may have been more sons than two and possibly their wives.
But Lot--though a righteous man--had no respect in his household. He had no great Godly example that had impressed his household. He apparently had adopted the thinking of the surrounding culture before separating from Abraham. Instead of seeking God as to where to go, as Abraham continually did, he chose what the culture around him prized: a good situation to maintain and, likely, increase his wealth. No thought apparently was given to the spiritual consequences of living in such a place. And though he continually saw the evil of this place and saw its influences upon his family, he did nothing to change the situation.
Could Lot have bowed to Uncle Abraham, yielding his wealth and his bickering servants to come as a son beneath him? Could he have chosen to become a part of what God was doing in setting the foundation for Israel?
Lot lost all respect in his family by his actions. He lost all but two wicked daughters who raised up future enemies of Israel. And two cities were destroyed. The angels could not find ten people in his household or his society that he had led into righteousness.