When Obi Wan kills Darth Maul, he uses his master's lightsaber to defeat the Sith Lord. After that, the lightsaber's fate is unknown according to this answer. Anakin becomes Obi Wan's Apprentice after the death of Qui-Gon Jinn. When Anakin is now a Jedi Padawan, he has constructed his first lightsaber. Now here's my question, Why didn't Obi Wan just gives his master's lightsaber to Anakin?
Why would he?
In the clone wars episodes The Gathering and A Test of Strength we see that lightsaber construction is an expected part of a Jedi youngling's training, and that each uses a crystal that is uniquely suited to them. One would assume then that there are advantages in using this crystal instead of just any old crystal.
The selection of the crystal also appears to be a part of the training / a rite of passage for the younglings, so I doubt Anakin would have avoided this. Thus if he had to make his own, and his own was likely better, or at least more personalised for him, why would he need to use Qui-gon's?
At least in Legends, it was a Jedi tradition to travel to Ilum and take part in a ritual called the Gathering. During the days of the Jedi Order, lightsabers were not generally inherited from a Jedi to Jedi, but initiates had to find the crystal that "spoke to them", and then construct their own unique weapon. Like Obi-Wan Kenobi said:
This weapon is your life!
Interestingly, Kenobi himself wanted to use his master's lightsaber for some years, due to losing his own in a duel against Darth Maul. That's also another obvious reason why he didn't give it to Anakin.