Improving weapons is certainly a thing.
There's a lot of precedent in D&D for this, from enhancing weapons to finding legendary lost swords with fantastic properties.
But you should worry about "bounded accuracy."
There's a design principle in 5e (whose documentation is nigh-impossible to find) called "bounded accuracy" which, briefly summarized, says "don't let the numbers get too big."
How big is too big? For your case--a new GM with new players--the easiest thing is to take a survey of the existing ways to get more damage out of a weapon and use that to calibrate. For a few examples:
- A rogue's signature combat mechanic is to add xd6 to their damage, once per turn. So your enhancement shouldn't get nearly that large, as it'd step on another class's toes way too much. (Another way of thinking about it: whatever cost your character pays for this enhancement, it's nowhere near the "cost" of having chosen a different class.)
- A paladin gets to slap extra d8s of damage on to attacks, at the cost of a spell slot. That's a pretty high cost, given that class doesn't get many slots.
- Magic weapon requires a second-level spell slot, concentration, only lasts an hour, and only gives a +1 to damage and hits. (Now I'd argue that most of that benefit comes from the hit, so there's a bit of muddying-of-going on here, but I think it's still worth looking at.)
- Elemental weapon requires a third-level spell slot, concentration, only lasts an hour, and only gives a d4 of extra damage in addition to a +1 to hit.
- A +1 weapon is uncommon, a +2 weapon is rare, and a +3 weapon is very rare. (Though again I'll stress, I believe most of that value comes in the hit-bonus.)
Hopefully you can see from the above that adding a d8 to a weapon's damage is a Big Deal. I'll go so far as to say going from 1d12 to 2d8, functionally adding 2.625 to a weapon's base damage is kind of a lot. (That's got crits baked into the differential, btw.)
You want to reward your players and add flavor to your game: that's great. I strongly suggest you find ways to limit it. A d8 to the weapon's damage, but only once per day and as a concentration-requiring effect, for instance. Or consumable charges on the weapon are a typical way to do this.
(The other nice thing there is that it gives your player ways to be tactical in their choices. If their weapon deals more damage then that's it, all said-and-done. But if they can choose to deal a little more damage, that's where the fun starts!)
The basic premise behind the bounded accuracy system is simple: we make no assumptions on the DM’s side of the game that the player’s attack and spell accuracy, or their defenses, increase as a result of gaining levels. Instead, we represent the difference in characters of various levels primarily through their hit points, the amount of damage they deal, and the various new abilities they have gained. Characters can fight tougher monsters not because they can finally hit them, but because their damage is sufficient to take a significant chunk out of the monster’s hit points; likewise, the character can now stand up to a few hits from that monster without being killed easily, thanks to the character’s increased hit points. Furthermore, gaining levels grants the characters new capabilities, which go much farther toward making your character feel different than simple numerical increases." Rodney Thompson, Legends and Lore, quoted in "Under the Hood--Bounded Accuracy" at roleplayeschronicle.com. Wizards.com has apparently lost many recent L&L articles, and any references to them seem to be broken links.
Notice what happens if you double your players' ability to dish out damage before their toughness doubles: you can't throw higher-level monsters at them because they don't have the toughness to stand with those foes, but their damage output makes level-appropriate monsters a cake-walk =(