I admit I cut my teeth on Traveller (Classic thank you) and AD&D and in that long ago time there were two systems for representing damage: hit points and stat reduction. Hit points were for those high fantasy games where you could get struck by dragon breath and live if you had enough experience.
Traveller was more grounded in reality (we knew this because it had math formulas for space travel that required a square root calculator!) You got hit and it hurt. Period. You were injured and impaired and you never got much better at taking damage (usually you got worse as age took its toll.)
In the games wth elves you got better at taking damage and it never slowed you down at all! this seemed counter intuitive to me (at least after I started playing Traveller). A ht with a battle axe would do more than shave a few hit points. Those suckers must hurt.
It was years later I heard Ernie Gygax explain what hit points represented: luck, skill, destiny if you will in avoided severe injury. When your hitpoints ran out so did your luck; roll up a new meat shield.
Now at first the world's most popular roleplaying game did not differentiate weapons damage. Everything did a d6. Okay a really good backstab could do 2d6. But most of the time a swipe was good for 3.5 points of damage. This also seemed counter intuitive to me. A battle axe did as much damage as a broadsword as a dagger as a sharpened stick (or nearly so, there were some pluses that crept in). But if you are talking about hit points being luck, skill, destiny and such then it makes sense. You're about as lucky avoiding a dagger thrust as a battle axe swing. Either can do for you. In Traveller terms this would be like firearms doing 1d6 or 2d6 (body pistol territory) and I guess you'd get 1d6 hit points per term in the military or whatever if you want to create a form of Traveller that used hit points (which I do not).
So yes in White Star a stick does 1d6 and an energy blade does 1d6+2. Avoiding getting your head smashed by a club is the same kind of luck as avoiding being decapitated by an energy blade. Dead is dead.
I do like some variation in weapons besides the story related ones. Obviously a dagger is not the same as a battle axe. You could jam a door open with a dagger, wedging it between the threshold and the door. It's be hard with a battle axe. You could use a spear to poke further into a dark hole than a sword etc. It just when you're laying out a fight for your players they might like some options and tactics.
Variable damage was done to death already. I'd like to suggest a critical hit system. Weapons could belong to a family according to their effects. Light weapons: pistols, one-handed melee weapons etc would score a hit on a natural 20, heavier weapons on a 19 or 20. If you use autohits on 20 I would say this supersedes the critical hits. In other words if the only way you could get a hit would be through rolling a natural 20 then you would not in addition get a critical hit as well. Another option would be to roll again with another successful hit indicating you've earned a critical hit. So for example if you hit on a 15 or better and rolled a 20 you would hit and roll again. On a 15 or higher you would get your critical hit. On a 14 or less you would earn a normal hit.
If you want to take the target's level into account instead of the double roll system you could allow them a saving throw to avoid a critical hit. You could also do all of this in Traveller style games, just replace saving throws with rolling under the appropriate stat on 2d6 or 3d6 depending on how lethal your referee is.
Some possible effects of critical hits follow:
1. Target is blinded for 1-3 turns (-4 on all attacks, maximum of close range ST may apply)
2. Target takes double damage (roll damage twice and add)
3. Target takes maximum damage (don't roll at all)
4. Target takes best of two damage rolls (roll twice and keep the highest roll)
5. Target's AC is made worse by 2 points (or they take double damage if unarmored), space suits are breached!
6. Target is stunned for 1-3 turns (may not attack, is +2 to be hit
7. Target is bleeding 1-6 hits per turn until first aid is rendered
8. Target is on fire and must make a saving throw to put it out or roll 1-3 damage for 1-6 turns
9. Target is knocked down and must get back up next turn
10. Target must make a saving throw or have a limb amputated. Double maximum damage and knocked out of the fight.
So you might have weapons families like Blunt, Edged, Slug, Laser and Blaster.
The GM rules that Blunt weapons have effect 9 unless the target makes a saving throw. Slugs (revolvers, rifles, SMGs) have effect 3 (maxed damage) if they succeed on a second hit roll. Lasers have effect 1 unless the target makes a saving throw. Blaster weapons have effect 5 unless the target makes a saving throw. He also creates a mono edged blade for a rare weapon type that has effect 10 but rules that not only must the user roll to hit again on getting a 20, the target gets a saving throw to avoid the effect.