All other spells command you to 'make a ranged spell attack' Magic Stone specifically gives the familiar permission, overriding the general restriction.
Given the 'specific beats general rule', if we can establish that the spell gives permission and that we consider giving such permission as a 'specific' situation, we can conclude that a familiar can attack a pebble touched by Magic Stone.
Permission over forgiveness:
Reading the spell text closely, you can see that it differs from all other ranged spell attacks:
You or someone else can make a ranged spell attack with one of the pebbles by throwing it or hurling it with a sling.
Compare to Fire Bolt:
You hurl a mote of fire at a creature or object within range. Make a ranged spell attack against the target.
You create a shard of ice and fling it at one creature within range. Make a ranged spell attack against the target.
The last two use the imperative mood, such that make is a command. Commands do imply permission.
Now, the two previous examples (of which many more exist) are instantaneous, you say! "The spell description is part of the action. But Magic Stone has a duration of a minute, it wouldn't make sense if Magic Stone used the imperative to say 'Make a ranged spell attack'" you protest.
Let's take a look at the closest comparison to Magic Stone, Produce Flame:
A flickering flame appears in your hand.
You can also attack with the flame, although doing so ends the spell. When you cast this spell, or as an action on a later turn, you can hurl the flame at a creature within 30 feet of you. Make a ranged spell attack.
This one does seem to give permission as well because previously the spell only describes how the flame glows. It might be as accurate to infer that 'can' here has the meaning 'it is possible to' not 'you are allowed to'. But after describing the secondary effect (the ability to attack as an action), and describing how (hurling), it still uses the imperative: Make a ranged spell attack.
In both Magic Stone, and Produce Flame permission/the ability has to be given, as it's not possible to make a ranged spell attack with a mundane pebble. If it merely made your ranged weapon attacks with said pebble be considered spell attacks, it could have been more clearly worded, similar to Fire Bolt and Ice Knife.
Each example follows the pattern:
Description of spell effects.
Command (i.e. 'make a ranged spell attack')
Let's circle back to see if Magic Stone follows this pattern:
Description of spell effect.
Permission (i.e. 'can make a ranged spell attack')
Condition ('by throwing it or hurling it with a sling.')
Can a familiar regularly make a ranged spell attack? No. If somehow they had access to cast one of these other spells (e.g. Fire Bolt), it wouldn't work because the spell commanding them to make a ranged spell attack doesn't grant permission.
In contrast, Magic Stone, via the properties of the rock or otherwise, says they 'can make a ranged spell attack'. Now, to do so, they have to meet a condition 'by throwing or hurling [the pebble] with a sling', something the familiar is permitted to do (so long as it is not an attack).
Are there any comparable times when a familiar is specifically permitted to attack? Yes, the Pact of the Chain provides this example:
Additionally, when you take the Attack action, you can forgo one of your own attacks to allow your familiar to make one attack with its reaction.
Here the conditional clause 'when <condition>, you can <auxiliary condition>' comes first, followed by the permissive clause, which is 'to allow your familiar to'. However, the meaning would be the same if we put the permissive clause first:
Additionally, your familiar can make one attack with its reaction when you take the Attack action and forgo one of your own attacks.
You can take it further and rewrite the sentence swapping the two conditions, preserving meaning:
Additionally, your familiar can make one attack with its reaction, by you forgoing one of your own attacks when you take the Attack action.
This statement now exactly matches the structure of the Magic Stone spell:
Permission (i.e. 'can make an attack with its reaction')
Condition (i.e. 'you forgoing one of your own attacks, when you take the Attack action')
Wrapping it up, like a well-made burrito
Now we've established that a) the spell grants permission, b) granting a familiar permission to attack is possible, we now have to consider if this situation is specific 'enough' to override the general restriction on familiars.
In the Pact of the Chain example, the familiar gains this specific permission through the warlock's feature. In the Magic Stone example, the 'someone else' (which can be any familiar), is granted this specific permission only when 'throwing [the pebble] or hurling it with a sling'.
Now the question boils down to: is being the familiar of a Pact of the Chain warlock more or less specific than being the wielder of the pebble? I'd argue while comparing them is difficult because the specificity of one is by association (with the warlock) and the other by circumstance (of wielding the pebble) — they are both specific enough to beat the general rule that familiars can't make attacks.
In conclusion, it seems possible that familiars can attack with a pebble imbued by Magic Stone because the spell explicitly grants that permission — this specific case overrides the general rule that would prohibit them from attacking.