120 feet (probably)
D&D 5e has two different kinds of range. Weapon range (expressed as short range/maximum range - PHB 147), and spell range (expressed as a single value - PHB 202-203 and in spell descriptions).
A thrown stone, as an improvised weapon (PHB 147-148) has the weapon range of 20/60.
A Magic Stone, as per the spell's description has a spell range of 60 feet. So we can infer that the spell range of a thrown Magic Stone equals the maximum range of a normal thrown stone.
If we extend this to using the Magic Stone as ammunition for a sling, it would seem reasonable to apply the same principle, and say that the sling stone's maximum range of 120 feet is the spell range of a Magic Stone thrown from a sling.
I say probably as this is an extension of the principles of the RAW, but as far as I know this principle is not spelled out in the currently-published rules.