Note: These were written with 5E in mind.
- Powerful. These are the simplest of magic weapons that just have a damage die that is one step up, so a shortsword that does 1d8 damage instead of the standard 1d6, or a longsword that does 1d10 damage for example.
- Brutal. Exploding damage. When you roll the max value on the damage die, you roll again and add the result. For a stronger variant, use 2d6 for damage and roll again whenever doubles are rolled (Doubles add and roll over [DARO] from Tunnels & Trolls).
- of Accuracy. Attacks are rolled with a d24. Advantaged attacks are d24+d20 and take the better result. These can introduce some issues, as criticals and fumbles are less likely. This should only matter if you are OCD like me.
- Bane. Advantage against a certain type of enemy (goblins, giants, undead, dragons, lawfuls, whatever). There are some more powerful variants that additionally do double or even triple damage to the type of creature they are designed against.
- Ultralight. These are the heavier one-handed weapons that deal 1d8 damage, like a longsword or a morningstar. They have been enchanted to weigh half as much as normal, and as a result they have the [light] and [finesse] properties, so they can be enjoyed by rogues and dual-wielders.
- of Wizardry. Enables the attuned wielder to use an action to cast a wizard cantrip at will. The most common versions have the Light spell, which is highly valued by humans and halflings. The Sword Coast Adventurer's Guide has a few cantrips that would work well with this, like Booming Blade or Green-Flame Blade.
I'd love to hear any other simple ideas in the comments...