While the various Star Wars movies and shows have shown the Jedi and their lightsabers extensively, Star Wars: The High Republic offers several new ways of using them. The main weapon of the Jedi Order, lightsabers were described by Obi-Wan Kenobi as being from a "more civilized age." Such an age is now being explored in The High Republic, a transmedia initiative set 200 years before Star Wars: Episode I - The Phantom Menace, a point in the timeline when the Jedi Order was at its peak.
The first book in the series, Charles Soule's Light of the Jedi, certainly delivers on this front. This is a version of the galaxy filled with Jedi, all of whom are capable of incredible Force powers that surpass those seen from the Jedi in the prequels, and they're also characters who inspire a great deal of hope. This difference extends to the lightsabers, too: as is to be expected from a story with so many Jedi, there's plenty of scope to explore their trust weapons, giving new ways and reasons they use them.
The new use of a lightsaber primarily comes from Avar Kriss, one of the Jedi Masters during the High Republic, and the central figure of Light of the Jedi. The green-bladed lightsaber, which has a crossguard hilt (but not one that is actually part of the blade, like Kylo Ren's), is not just her weapon, but, as described in her own thoughts, "an instrument" that she uses to connect with the Force. Avar is adept at seeing the natural bonds in the Force, which she hears as a song, connecting with other Jedi and being able to focus in on certain aspects. She uses the Force to make her lightsaber spin in the air, and in turn the noise it produces as it moves faster and faster becomes the song of the Force. As described in Light of the Jedi:
The weapon began to rotate, slowly, like the blade of a windwill. It made a sound as it moved through the air, a low, droning hum. Avar breathed - in, out - and the blade slowly sped up. The tone of its passage through the air changed, no longer a low drone but a higher pitch, a lovely round note. The lightsaber moved faster, its blade now too fast to see; a green circle of light with a shining metallic center. It was beautiful, but Avar closed her eyes. She did not need to see. She needed to hear. Her lightsaber was not just a weapon. Here, now... it was an instrument."
Jedi have long had a connection to the lightsabers beyond them just being weapons - each Jedi bonds with the kyber crystal inside, which is what gives a lightsaber its color, specifically fitting the individual - but this "sabersong" makes that much deeper, presenting a Jedi who is far more attuned to her lightsaber. Avar and her saber come together with the Force, all three working together in perfect harmony.
The High Republic also introduces other ways for the Jedi to use their lightsabers. The Knights and Masters of the era fly starships called Vectors, which need a "lightsaber key" in order to work. This means that they can only be piloted by a Jedi, who has to use their lightsaber to get it working, but it's also designed to ensure that they themselves respect the Vectors and are considered when flying them. They must use the ship's weapons system with the same discipline they would their lightsabers, which in turn further shows just how much care and thought they need for even wielding their sabers in the first place.
Light of the Jedi isn't the only story in The High Republic to add new layers to Jedi and their lightsabers in Star Wars. Justina Ireland's novel A Test of Courage also brings back the lightsaber whip, which was part of the old Expanded Universe (now Legends). The lightwhip is much as it sounds - like a lightsaber except, instead of a straight blade, it's flexible and can be used like, well, a whip. In Legends, lightwhips were their own specific weapon, separate from lightsabers, but The High Republic adds a new twist: now, a regular lightsaber can be modified to be made into a lightsaber whip, switching from sword to whip and back with a switch.
While present in the High Republic, such weapons weren't common among the Jedi, which perhaps explains why they're not seen in any of the Star Wars movies. They are also more commonly used by the Nightsisters of Dathomir, meaning they're a weapon with more dark side links, which would again explain why the Jedi don't use them, alongside the fact the blades aren't as strong as that of a regular lightsaber. The High Republic also includes another more dark side weapon: a Sith lightsaber, called a lightspear. from which the Jedi Jora Malli takes a kyber crystal for her lightsaber. With new techniques, Force connections, and even more links to the dark side, Star Wars is still finding new, interesting ways of exploring the Jedi and their lightsabers.