What’s the Elderberry Tales?
With the stipulation that we’re using the 5e mechanics, I assume simply house ruling slings to have a higher damage die is off the table, yeah? Just checking cause that would obviously be an easy way to make it more viable. Without that, the main tools we have to make slings useful are its other properties that stand out from other ranged options. It’s the only ranged weapon to deal bludgeoning damage, so if bludgeoning damage is highly desirable in your campaign, that will inherently make slings an appealing ranged option. Perhaps skeletons are a very common threat in your campaign. Or, if you bring back the ahistorical but commonly used in fantasy notion of clergyfolk not being permitted to use weapons that draw blood (i.e. deal piercing or slashing damage), then slings would be the only ranged weapons usable by clerics and perhaps paladins.
Slings are one of a very small number of ranged weapons that lack the two-handed property, the only others being thrown weapons, and the martial hand crossbow (and pistol if your game has firearms). You still need a free hand to load it, but… I don’t know, seems like there could be some way to take advantage of that…
Slings are very cheap. If money is a very significant concern in your campaign, a sling might become a more appealing option, especially if you allow using rocks instead of proper sling bullets as ammunition. Maybe the sling counts as an improvised weapon when used that way, so there’s still a reason to buy ammunition if you can afford it.
And, of course, you can make slings more appealing with their ammunition. Perhaps magic sling bullets are extremely abundant and easily available. In-universe, I could see there being a common folk practice of carving runes and/or holy scripture into sling bullets, to justify the commonality of such magic items.
EDIT: Oh, yeah, and there’s the magic stone spell. That’s probably not as much of a draw in the 2024 rules due to the new and improved version of True Strike, but still. It’s something.