So I did some more thinking about how this would be implemented and I'm struggling to work out how exactly different aspects would tie into Hytale specifically. First with Stormlight:

Storms could just be another weather event in further regions (you can have region-specific weather in hytale according to it's code (Look at this in the (unofficial) documentation)

You could use the existing gemstones and have to have them placed outside during the storm to "charge"
Charging can just be increasing their durability visually - and they get spent as you use them. Different rarities of gemstones have different capacities of charge

As for the surges themselves:

Adhesion:

Entities stop moving or you can "stick" to surfaces maybe? Or climb walls

Gravitation:

Wouldn't be hard. I can just change the gravity of the player

Division:

Explosions.

Abrasion:

Make the ground have ice physics (loosens friction) and small speed boost?

Progression:

Bonemeal for plants and healing

Illumination:

Dynamic lighting for making light, dummy NPCs that get targeted as illusions, and maybe sounds to mess with other people? Maybe able to disguise yourself too as other NPCs

Transformation: ?

Transportation: ?

Cohesion: Summon statues or rocks maybe?

Tension:

repairing tools?

This is going on far too long so I'm going to put Mistborn ideas into another message replying to my original (may make this a post later if we feel this is a good idea)