Xjyuk

There appear to be two plausible interpretations.

1. The magic stone’s effect amplifies and thereby replaces the regular damage of the stone hurled by the sling.


2. The sling is hurling the stone at high velocity causing regular damage and the spells effect is that it causes it’s damage (additional).

Question

What damage dice are rolled when a creature is hit by a magic stone that was slung by a sling, and why?

Example: A Druid named Agar has +3 to his spellcasting ability modifier, he carries a sling and has already cast magic stones on a few pebble carried in his pocket.
In the distance, a goblin slowly crests a hill and comes into view.
Agar readies his sling as the goblin advances from about 100 feet away.
Agar slings a magic stone toward the goblin.
Before the goblin can react, he is struck by the stone.

Does the Goblin take 1d6+3 damage?

Does the Goblin take 1d4+1d6+3 damage?

Magic Stone

You touch one to three pebbles and imbue them with magic. You or
someone else can make a ranged spell attack with one of the pebbles by
throwing it or hurling it with a sling. If thrown, it has a range of
60 feet. If someone else attacks with the pebble, that attacker adds
your spellcasting ability modifier, not the attacker’s, to the attack
roll. On a hit, the target takes bludgeoning damage equal to 1d6 +
your spellcasting ability modifier. Hit or miss, the spell then ends
on the stone. If you cast this spell again, the spell ends early on
any pebbles still affected by it.

Sling

Sling, damage 1d4 blugeoning (PH, 149)

The magic stone always deals the same damage regardless of launching method

The relevant parts of the spell text are:

You or someone else can make a ranged spell attack with one of the pebbles by throwing it or hurling it with a sling. [...] On a hit, the target takes bludgeoning damage equal to 1d6 + your spellcasting ability modifier.

The normal damage of the sling is irrelevant, because you are not using it to make a weapon attack. Instead, you are making a ranged spell attack, which deals damage as described in the spell.

In your example, the damage will be 1d6+3.

What if I'm using a +1 magic sling?

A +1 weapon states:

You have a +1 bonus to attack and damage rolls made with this magic weapon.

Since this text doesn't limit the bonus to only weapon attacks, this +1 bonus applies to both the ranged spell attack roll and damage roll of a magic stone launched using the sling.

What if the target is more than 30 feet away?

The rules for the range weapon property say:

Range. A weapon that can be used to make a ranged attack has a range in parentheses after the ammunition or thrown property. The range lists two numbers. The first is the weapon's normal range in feet, and the second indicates the weapon's long range. When attacking a target beyond normal range, you have disadvantage on the attack roll. You can't attack a target beyond the weapon's long range.

As with the +1 weapon case, the rule for weapon range does not limit itself to only weapon attacks, so the disadvantage beyond the sling's normal range also applies when using the sling to make a ranged spell attack with the magic stone.

The same principles can be applied to any other questions about which properties of the sling apply to the magic stone attack and which ones don't: if the property applies only to weapon attacks, it is not relevant, but if it applies to all attacks made with the sling, it applies to the magic stone attack.

RAW

Slings are a Simple Ranged Weapon (PHB 149) thus allowing a ranged attack. While using the magic stone in a sling after the spell been cast is considered a ranged spell attack (Elemental Evil Player's Companion , pg. 160).

You touch one to three pebbles and imbue them with magic. You or
someone else can make a ranged spell attack with one of the pebbles by
throwing it or hurling it with a sling

With D&D 5th edition rule that specific beats general (page 7). I would rule that the 1d6+spellcaster ability modifier would substitute for the normal sling damage as the normal ranged attack is converted to a ranged spell attack per the spell's description

Commentary

It would be a trade off using a sling versus being thrown. As the sling attack would be made at a disadvantage after 30 feet. While with throwing the magic stone by hand will allow a ranged spell attack out to 60 feet.

If thrown, it has a range of 60 feet.

The advantage of using a magic stone with a sling over throwing is to allow an attack with a disadvantaged ranged spell attack roll out to 120 feet.

1d6+3

As the spell states:

You... can make a ranged spell attack with one of the pebbles by... hurling it with a sling

When Agar performs the Readied action, he is performing that spell attack, and follows the rules that the spell describe:

On a hit, the target takes bludgeoning damage equal to 1d6 + your spellcasting ability modifier.