What does Genesis 19:20 mean?

Over the last few hours, the Lord has shown great mercy to Lot and his family. The angels have saved them from an angry and rapacious mob in Sodom, have physically removed them from the doomed city, and have now told them to run for the hills without stopping to save their very lives. At every step along the way, these angels have acted with extraordinary grace and patience with Lot and his family.

Despite all of this, Lot rejects their commands to run into the hills. He apparently does not believe they will make it in time. Now Lot asks if the Lord would maybe allow them to flee to a small city nearby. This is not only a request that he seek shelter in a closer place, but it also implies God withholding judgment on that tiny town, as well. The implication in the request is that this little village was likely going to be destroyed if Sodom was annihilated.

Lot has been saved, along with his family, from the judgment of God on a wicked city which he called home for many years. This makes Lot's request seem extremely bold. Abraham had asked the Lord to spare Sodom if as many as ten righteous people could be found—which, as it turned out, was more than Sodom could muster. Lot asks God not to destroy this presumably wicked but much smaller city, simply so he and his family can shelter there.

Neither the Lord nor the angels rebuke Lot for this request, as we'll see in the following verse.