While only time will tell how much of the series will feature Anakin's lightsaber, it is clear that Obi-Wan didn't have a personal gain on his mind when he reached for Anakin's lightsaber at the end of Star War: Episode III - Revenge of the Sith. In The Clone Wars, and at the beginning of Revenge of the Sith, he was already using his new lightsaber, the very one that would later be wielded by Luke and Rey. Obi-Wan took the lightsaber because he wanted to show respect for the friend he once had before he was corrupted by the Sith. While Obi-Wan's new Disney+ show Obi-Wan Kenobi, which will take place on Tatooine after Revenge of the Sith, is likely to shed more light on the dormant years of the lightsaber's lifespan.
It is clear from Obi-Wan's dialogue that he kept the lightsaber because of the respect he had for it. The Rise of Skywalker implied a connection between a saber and its Jedi wielder by showing Rey performing a burial of that saber. The Disney+ series Obi-Wan Kenobi has the chance to address these intentions, as the second trailer featured an argument between Ben and Owen regarding training Luke in the ways of the Force. Being a Jedi in the Star Wars universe draws obvious comparisons to real-world religions, and a Jedi's saber has the same spiritual reverence and symbolic importance that is given to many sacred objects throughout our various faiths. He showed his admiration for the ideals of the Republic era, and what the Jedi once upheld, by explaining that they were used by the guardians of peace and justice.
It is clear to Obi-Wan that an object of this importance wasn't meant to be left out of sight. In Star War: Episode III - Revenge of the Sith, Obi-Wan took the lightsaber of his fallen apprentice.
During the age of the Republic, lightsabers were often made by their owners.
It may have crossed Obi-Wan's mind that he could pass on the baby to Anakin's family. According to canon, Obi-Wan stole the weapon from Anakin after their fight. He believed that a Jedi's lightsaber was a symbol of peace and that it would carry the legacy of its wielder's generations. There is a lot of context that can be retrieved from Ben Kenobi during the main saga that helps answer the question of why he took Anakin's saber. While singing the praises of the lightsaber, Kenobi dismissed the clumsy nature of a blaster.
The part of the wielder that was good was preserved since the saber had not yet been bled. Obi-Wan made sure to get the lost saber when he lost his lightsaber. It is likely that he would have created a new saber after the battle. The crystals that were used to make the lightsaber were found in the Crystal Caves. He sternly remarked, "This weapon is your life." He said the lightsaber was an elegant weapon for a civilized age.