After Yoda's death in 4 ABY, the lightsaber with its accompanying storage box ended up in the possession of Yoda's last Jedi apprentice, Luke Skywalker. In 9 ABY, Skywalker offered the saber to Grogu, a young member of Yoda's species; who was being trained by Skywalker at his Jedi academy on the Outer Rim planet Ossus.
This brings us to the other explanation for Luke having Yoda's little saber in Boba Fett: Yoda built a second lightsaber after his duel with Palpatine and kept it hidden on Dagobah. Then, after Yoda gave up the Force ghost, Luke would have gone through his stuff and found it.
After Yoda's death in 4 ABY, that lightsaber ended up in the possession of Yoda's last Jedi apprentice, Luke Skywalker.
Grogu chose the latter, meaning he gave up his chance at getting his own lightsaber. After picking Din's gift of beskar chainmail armor, Grogu left the other option of Yoda's lightsaber in the hands of Luke.
After losing his father's lightsaber on Cloud City, Luke Skywalker constructed a replacement with a green plasma blade, its hilt similar to that of Obi-Wan Kenobi's weapon.
The weapon had a kyber crystal at its heart, and emitted a yellow-colored plasma blade when ignited. The emitter matrix on Rey's weapon featured a rotating gear-like mechanism, which caused the emitter of the lightsaber to open up and allow the plasma blade through.
Luke Skywalker's lightsaber was changed from blue to green when they were filming because it was difficult to see. This became apparent during act one of Return of the Jedi when the battle on Jabba the Hutt's sail barge was shot during the day. The blue lightsaber blended in with the clear blue sky.
Grogu will likely never become a real Jedi. After being given an impossible choice by Luke Skywalker in Boba Fett Episode 6, Grogu chooses to leave Yoda's lightsaber behind and join his surrogate father Mando (aka Din Djarin) once again.
Yoda kept a second lightsaber on Dagobah inside his hut in 3 ABY after his previous one was lost during a duel in the Galactic Senate and then incinerated during a celebration of the Jedi Order's eradication in 19 BBY. Luke Skywalker eventually had the lightsaber in his possession after Yoda's death in 4 ABY.
In truth, the Sith normally wielded enough powers to fight without a lightsaber, but every apprentice still learned to use one as part of his or her training. Sidious himself felt that he and his peers had outgrown the use of lightsabers, and only continued carrying them to mock the Jedi.
Yoda held the title of Grand Master of the Jedi Order since the days of the High Republic Era. At some point, Yoda took on a human named Kantam Sy as his Padawan. The two were stationed at the Jedi Temple on the planet Endovar.
Legend had it that Yoda—a Jedi who became Grand Master—was trained by N'Kata Del Gormo. A Force-sensitive Hysalrian, N'Kata Del Gormo was trained in the ways of the Force and achieved the rank of Master within the Jedi Order.
When Obi-Wan tells Yoda that Luke is their “last hope,” Yoda says, “No. There is another.” While the events of Return of the Jedi suggest this “other” person was Leia, it's also possible Yoda was referring to Rey.
Yoda : [Deleted Scene - A dying Yoda confesses to Luke about the truth about Darth Vader] Unfortunate that you must rush to face him. And in complete of your training. That no ready the burden with you.
These are presented by Luke as almost a choice between the Jedi Order and his surrogate father, Din Djarin also known as The Mandalorian. Previous to this, Star Wars canon told us that Yoda's lightsaber had been destroyed.
In other words, it was already established that Yoda had two lightsabers, and that there was one with him on Dagobah even after one was presumably destroyed by Sidious.
Yoda's lightsaber was the weapon carried by Grand Master Yoda. Its blade was shorter than that of a normal lightsaber to match Yoda's small stature, and was green. He used this blade to perfect all styles of lightsaber combat, most notably Form IV: Ataru.
Purple. Only one person in canon wields a purple blade: the Jedi Council's Mace Windu. He uses it to decapitate the deadly bounty hunter Jango Fett in Attack of the Clones and to deflect Darth Sidious' Force Lightning with it -- disfiguring his face -- in Revenge of the Sith.
Yoda too had to go on a voyage to find his Kyber Crystal. He engaged on his eventful journey to the ice-ridden caves of Ilum where he found his crystal. Once acquired, the onus was on him to take his crystal back to the Jedi Temple and have his lightsaber forged.
In the Star Wars Legends continuity, the founders of the Jedi Order were Jedi Masters Cala Brin, Garon Jard, Rajivari, and Ters Sendon.
R2-D2 is confirmed to have been on Coruscant at the time of Order 66, when he was seen on Skywalker's ship. R2-D2 may have been familiar with Grogu through his allyship with the Jedi Order and gone to save him during Skywalker's massacre of Jedi younglings, which left him with a window to leave the ship.
Leia Organa's lightsaber was a blue-bladed lightsaber powered by a kyber crystal in the core of the hilt. It was silver and copper in color and featured mother-of-pearl inlays. Organa's weapon was considered a work of art, having an elegant symmetry with silver and gold hues that evoke to her upbringing on Alderaan.
It's possible that Rey adopted the yellow-bladed lightsaber (or perhaps the kyber crystal conjured a yellow blade based on the changes within its wielder) in honor of those emotionally detached sentinels tasked with guarding the Jedi Order.
The lightsaber's amethyst color was chosen by the actor portraying Windu, Samuel L. Jackson, who wanted a purple lightsaber because he was "the second baddest Jedi in the universe" and so he could find himself among the many Jedi featured in the Battle of Geonosis scene.