“Thus both the daughters of Lot were with child by their father. The firstborn bore a son, and called his name Moab; he is the father of the Moabites to this day. As for the younger, she also bore a son, and called his name Ben-ammi; he is the father of the sons of Ammon to this day.” Gen. 19:36-38
First we must realize it would not at all be impossible for the two sisters to get pregnant at nearly the same time. It is quite common knowledge that when a woman spends a good amount of time around another woman, their monthly cycles often align.
As for the meanings of the names, Moab is likely a wordplay on “from my father” (Plaut) and Ben-ammi means “son of my people” or “son of my kinsmen” which is a little less obvious. This incestuous origin would affect how the peoples of Moab and Ammon were viewed by the Israelites with regard to their sexual morality.
The Moabites and the Ammonites became somewhat a thorn in the side of Israel, though not quite to the same degree of some of their neighbors. In Judges 3 we see that the Moabites gathered with the Ammonites and fought against Israel and defeated them, and Israel served the king of Moab 18 years. However, Israel later defeated the Moabites. In Ezra 9 we see that the Moabites intermarry with the Israelites, of which God did not approve.
Interestingly, God allowed the Moabites and Ammonites to keep their land rather than give it to the Israelites. This may well be due to God’s promise to Abraham concerning the land, then, by extension, to Lot. We know God promised the land to Abraham; then in Genesis 13:9-11, Abraham suggests he and Lot separate due to the land required by their herds, and implies that Lot should go and take whatever land he sees fit as his own, which he does. As we see in these verses, Lot’s daughters sleep with him, and this ultimately gives rise to the peoples of Moab and Ammon. In Deuteronomy 2:9 God says to Moses “Do not harass Moab or contend with them in battle, for I will not give you any of their land for a possession, because I have given Ar to the sons of Lot for a possession.” and in Deut. 2:19 “…when you approach the frontier of the sons of Ammon, do not harass them or contend with them, for I will not give you any of the land of the sons of Ammon as a possession, because I have given it to the sons of Lot for a possession.”
Ultimately things do not end well for the Moabites and Ammonites however, according to Zephaniah 2:9:
“Therefore, as I live,” declares the LORD of hosts, The God of Israel, “Surely Moab will be like Sodom And the sons of Ammon like Gomorrah– A place possessed by nettles and salt pits, And a perpetual desolation. The remnant of My people will plunder them And the remainder of My nation will inherit them.”
The Moabites get an honorable mention however, as Ruth was a Moabitess and was ultimately an ancestor of Jesus the Messiah.