Knowing that they would most likely take pity on him and take him to raise him as their own. But Billy most likely saw Jake's parents, and placed the Finn some where near their path. Bill took the Infant Finn, maybe to raise him himself, but felt that he wouldn't be a good parent for the young Child.

At Billy's request to ensure the Lich stayed in their forever. After trapping the Lich in Amber, probably around the time when Bubblegum's ancestors were barely building the castle. Knowing that Billy could never see bring himself to hurt an infant. Instead of fighting within in his home, he decided to let himself be frozen in the amber. Billy may have discovered The Lich's home, and knowing Billy would take away from The Lich's grasp. As cuts and broken bones seems to heal a bit faster on Finn. As Finn has been in situations that would have most likely killed a normal Human being. The Lich probably gave Finn more endurance, to survive Ooo's hostile environment. Since infants are considered the purest of hearts and The Lich was unable to posses his body at that time.

One he could either raise as his own so he may continue the Lich's work or a new host body he could inhabit when it reached maturity.

The Lich most likely created finn, using Necromancy and other dark magic, was able to create an infant Human. So it would make sense to try and create the most destructive creature in the land of Ooo, as they were also his creators. After battling Billy for so long and knowing this one day, Billy may succeed and eventually kill him, The lich would most likely try to create a new host body or at least an Heir to continue his work on destroying all life. It wouldn't be surprising if he tried in someway to ensure his immortality or at least make an heir to his dark powers and kingdom. Now reading the Lich's post and the theories that he is the last of his kind. Here is a theory a few friends and I have come up with.

Adventure time lich