Making it much lighter is pretty much needed unless you somehow make it so good it can be competetive with its current weight - which would mean a big overall decrease in ship speeds.
Making it fully recyclable in normal fabricators would also be good, or making it repairable when in spacedocks/bays; either of these would lower the requirements for a mothership that can maintain armored ships by a good bit.
Simply removing hex shields is another option; again, because this would force armor rather than just make it more appealing, it would lower both ship speeds and costs a good bit. It would be a major nerf to gunships like we had in the last game, which had just one or two hexes as protection, and so make small beam fighters more appealing than they are now.
You could give it one SP, allowing it to take small amounts of fire without any damage - the armor radiating back the absorbed heat or something - which would make it stronger on fighters and missiles.
Buffing the HP of the current piece, and adding a new armor hex "Light Armor" with less health, less cost, and a bunch less weight could work - the heavy version could be used for internal barriers like on my old battleships, and the lighter one for protection on fighters. This would also go well with the SP idea, if you want fighters to be stronger. These changes would also theoretically make protected missiles cheaper, but a Contour 1 currently is so good for missiles that I don't think we'd see much of a real change to them.
Edit: For a more wideranging set of changes, I like the idea of a protection nerf to shields, equivalent damage nerf to lasers, mass buff to armor, and cost buff to disruptors. This would ideally make disruptors more popular because the protection nerfs would mean they are no longer useless, laser nerf making them less outclassed, and the cost buff that they're no longer half the price of an entire fighter for a Medium. Shield/Laser interaction is good now, so the equality in strength reduction should keep that the same. Armor would become more popular because it would no longer slow ships down awfully, and the damage nerf to lasers and increased prevalence of disruptors would mean that shields are no longer so much better at protection (though they would need to be somewhat better.)
That would be a lot of delicate changes, though, and I haven't really thought out how it would interact with power, propulsion, or ship design.