Zadok_Allen: Personally I'd favor the lutes and instruments as seen in "For the King". Music is a fitting theme for magic, not least for all kinds of utility magic - including healing. Ofc that's only a "weapon" insofar as it goes into the weapon slot, but considering the off-the-beaten-tracks kind of the question I reckon that's only fair.
Besides that healing is ofc often associated with magic in games, so staves and wands are the no-brainer classic answer in my book. Now that I think about it I could also well imagine a flagellum with HP damage and purging effects, effectively healing debuffs, disease or poison for instance. Kind of a grim theme here, but that can be an advantage in a game with a fitting scenario. Although I wouldn't know an example it wouldn't surprise me if there already were multiple.
*As for the syringe: It exists in countless games, for instance Underrail. Problem is: It's a straight out consumable and not a weapon. It's fine for the healing part, but less of a weapon even than a lute.
So, something like the Wind Flute in Lufia: The Legend Returns?
You equip this flute in the weapon slot, and when you "attack" with it, it heals the party based off the user's current HP.
Probably not needed for the main game, but handy for later Ancient Cave trips. (You have to have reached level 100 of this 200 floor dungeon to get it, but that gives enough time for it to be useful, especially if you choose to start on floor 101.)
(By the way, it can actually be fun to enter that dungeon on floor 101 once it's unlocked; the enemies and resource management are a decent challenge for a level 1 party, and the Wind Flute is helpful there, at least until you get both Absorb and a decent healing spell.)
Lufia 1 has a Wind Flute, but it's a usable item that's like the Sage's Stone in Dragon Quest games; it heals the party by a small amount when used, but can be re-used infinitely (but only during combat).
Edit: Another good example is the Dream Harp in Final Fantasy 3 (3D Remake); it turns the Bard's Sing command into a weak party-wide heal that costs nothing to use and always goes first in the round.