I’m amased at how many answers get the meaning of inertia wrong, even while claiming others are wrong.
To be clear, inertia is a mass’s property that makes it require some effort to accelerate. For straight-line motion that is equivalent to mass and is the meaning of m in F=ma. For rotation, it depends on the distribution of mass around the center of rotation, and this moment of rotational inertia is noted as I. The equivilent of force = m × acceleration becomes torque = I × angular acceleration. In a swing, the rotation version might be important.
Besides the problems with the answers, it promps me to wonder just what did the OP mean? From his prose it is clear that the word was used correctly: he wants the sword to be easy to swing, yet still deliver blows like a weighty object.
Let’s start with angular inertia and torque, since that’s something that can in fact be altered. When you open a door you push at the edge as far from the hinge as you can, and push at 90° to the current position of the door. If you pushed an inch from the hinge it would require a lot more force to push it the same amount. An object’s I works the same way: imagine a wheel. If the mass is at the rim it will be harder to get moving than if the mass were concentrated at the hub.
So you can change the effort required to swing something by changing the shape and points of leverage.
Imagine the mass concentrated at the far end, more of a war hammer. If the weight could be retracted to the handle then swinging would be easy! You just lowered the value of I. Now you have this lightweight stick moving in an arc at some considerable speed, and you release the weight and send it back to the far end. What happens? The rotation slows to nearly nothing! You can’t deliver a hard blow after all.
Now for straight-line inertia (mass). If you had some way of turning the inertia off and on again it would play havoc with the laws of physics. Conservation of energy, conservation of momentum, and if you patch that up by delivering energy and recoil from a magical source in a simple way, you break relativity.
But the angular form shows how it can be made to work without breaking physics. Imagine this real-world implementation: the sword is hollow and filled with dense fluid. You swing it empty (light), and then the fluid is replaced. In the physical example you need hoses and pumps, and in principle you see that the fluid needs to catch up with the object and match velocity. This requires an engine (pump) to do so.
Now rather than physical fluid you can imagine the momentum being drained and refilled somehow, using more convienient wires (like electrical conductivity) and at higher speed. But the end result is the same: you will either lose the speed when you fill it back up with non-moving weight, or some external engine matches the speed and this takes as much energy (and picks up recoil) as it would have taken to accelerate the mass normally.
You can’t instantly turn the mass back on: it is a process which delivers the mass back to the state (position and velocity) you wanted in the first place.
So the real (useful) point is that the force is supplied by an external engine, not your muscles. This is what we do in the real world with hydrolics! You can turn your car’s steering wheel with no effort—the weight you feel is left on purpose for better control—and the tires follow your command, powered elsewhere.
Imagine a lightweight stick that you swing and point, and a hydrolic system moves the heavy sword to match. If the motion keeps up exactly, the stick could be hidden inside a small hollow in the sword which protrudes as a handle. It would feel like a weightless sword.
Now in the real world this would be an exoskelliton, or (as with construction machinery) not co-located with the control. You can imagine the magic making what’s essentially an invisible exoskelliton, using magic-provided energy to move the weight. And there you have it.
But why not the equivalent of magic-implemented buldozers, cranes, and end loaders instead? Why a sword when the technology could be used to make a projectile weapon, or simply toss the enemies aside?