Yes, although I would restrict it to pets that get an attack
Your question rests on the premise that a pet mouse would use the stat block for a rat. I don't agree with that underlying assumption, but will address that later. So to begin with, let's find common ground and assume that your druid has a pet rat.
Rats are size Tiny creatures of the Beast creature type and with an Intelligence of 2. They meet all qualifications for Beast Bond and Speak with Small Beasts, so no problems there. Your gnome druid makes some magic stones and hands one off to his little friend Templeton. What happens then?
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...Hit or miss, the spell then ends on the stone.
The interesting part of this description is that the stones can be used by you "or someone else". Does your little rat friend count as someone else?
This is very odd construction, not at all like other spells and abilities, which would instead say "You or a creature..."1. Indeed, the mechanics of the spell are otherwise very much like that of Freezing Sphere in that the projectile you make can be thrown or slung, or handed off to another creature to throw or sling. So it is odd that magic stone refers to this other being as "someone else" rather than "a creature".
While the game does not have an explicit definition of what is a creature, at least creatures are better defined than what qualifies as "someone". However, we do find references to "someone" in places like the Hiding sidebar ("When you hide, there’s a chance someone will notice you even if they aren't searching"), the Opportunity Attack rules ("...when someone or something moves you without using your movement..."), and the rules on Reactions ("on your turn or on someone else's"). Thus, I think we are on safe ground to say that 'someone' is roughly equivalent to 'another creature' and conclude that your rat qualifies as 'someone' who can receive the magic stones.
Can the rat throw them? Here we get to what Trish, in their answer, called: "Apply common sense first: Most animals lack the physical capabilities and the intelligence to throw stones." Does it break verisimilitude to have a natural rat tossing stones? Certainly a mouse or rat can deliberately pick up and drop a small stone that might fit in their hand and can easily be trained to do so, even without magic or the special communication abilities of gnomes. I find it harder to believe that one would deliberately toss said stone at something, and they might not even have the anatomy to make a proper throwing motion - but I don't actually think that is necessary.
The magic stone spell makes it clear that the target of the magic is the stones themselves. It takes normal pebbles and 'imbues them with magic' (until the spell ends 'on the stone'), such that someone may toss them, and if they do, they have your spell bonus to hit, not the thrower's. They also have no penalty for long range2. That to me justifies that if the rat is merely capable of grabbing and releasing the stone while contemplating a target, the magic on the stone itself will permit it to fly toward and strike the target. Personally, I would draw the line at slinging a stone, however. Even if you make Templeton a proportionally-sized sling, I just don't see him standing on two legs, whipping the sling about with one paw, and letting go with some but not all digits of the paw, which for me is what would be required to activate the magic of the stones. Your DM will have to decide to what extent the rat needs to interact with the pebble to activate the magic, and whether that interaction is realistic for the rat. But DM-permitting by all means equip your rat pet with magic stones and give them advantage on their tosses with beast bond.
When the rat receives the magic stone, it is permitted to make a ranged spell attack, even though such an attack is not noted in their stat block, where their only attack is one bite. It is clear that, again, it is the magic of the spell that is permitting this. Handing off a stone to your human fighter companion would similarly give them a ranged spell attack, even though they do not have the Spellcasting feature. To use the stone, though, I believe the rat needs to be capable of making an attack, that is, of taking the attack action. A rat normally spends its attack action on its bite (a melee natural weapon attack), but the spell is permitting it to instead spend its action on the ranged spell attack.
I do not think, however, that the spell is capable of providing an attack to a creature that otherwise does not get one. A shrieker, for example, does not get the Attack action and I do not think the spell can go so far as to give your fungus friend an attack. There has to be a line somewhere, both of verisimilitude (can you arm a cockroach with a magic stone? A housefly?) and of balance (with three mice does your first level druid get three attacks per turn?). And here, as we approach that line, is where we need to distinguish between rats and mice.
It's true that rats and mice are similar, and that we are enjoined to use the stat blocks of similar creatures when one does not exist for our target:
Monster Manual (317):
Other Animals. A book of this size can't contain statistics for every animal inhabiting your D&D campaign world. However, you can use the stat block of one animal to represent another easily enough. For example, you can use the panther statistics to represent a jaguar, the giant goat statistics to represent a buffalo, and the hawk statistics to represent a falcon.
However, in this case, a mouse is too different from a rat for me to comfortably use the stat block of the rat. A European field mouse, for example, weighs around 23g, while a Norway rat is some seven to more than twenty-times heavier3. A rat has just 1hp, can do 1 point of damage, and has a CR of 0. There is no way we can meaningfully express in a stat block something significantly smaller and more innocuous than a rat. Things that are too small or harmless to have their own stat block are not typically treated as creatures. As NautArch rightfully points out, this design principle may not be explicitly stated. However, the intent becomes clear when we look at a number of related game features.
Some animals are represented only as swarms
There are rats (CR0), and there are swarms of rats (CR1/4). However, there are many swarms for which individual creatures are not given stat blocks. There are (MM338) swarms of insects, beetles (and scarabs, and hoard scarabs), centipedes, spiders, and wasps. However, of these, only spiders are treated as creatures, and we do not have a stat block for an individual insect, beetle (or scarab or hoard scarab), centipede, or wasp. We do not have a stat block for an solitary rot grub or a lone maggot, despite the fact that they can be treated as monsters when they swarm.
Can it be kept on your person?
Where do you keep this pet mouse? If it is a cute but harmless narrative feature, and it can hide in your pocket during combat, it is not big enough to get its own attacks. A creature, by definition, cannot share its space with you, as NautArch points out. Even a tiny creature 'controls' a 2.5 x 2.5' space in combat, and cannot willingly end its turn in your space, nor you in its. Narratively, you might be able to carry a rat on your person when things are calm, but during combat it is going to get in your way, and vice versa. If you can keep the mouse in your pocket when blades are flashing and spells are being slung, by definition it is not a creature.
Normal Plants are Treated as Objects
In its discussion of monster / creature types, the Monster Manual (7, 8) says:
Plants in this context are vegetable creatures, not ordinary flora. Most of them are ambulatory, and some are carnivorous.
'Ordinary flora', by definition, do not have a creature type - and thus might not be considered creatures. Are they alive? Of course. But the game rules do not treat them as creatures, because the PC's will not be interacting with them in combat. As the MM says, a monster "is defined as any creature that can be interacted with and potentially fought and killed." Things such as normal plants, and mice, are not normally interacted with in this way. They can be, but usually are not. Thomas Markov expresses this principle as that tiny harmless entities "should be generally ignored until someone intentionally engages with them, at which point they are treated similarly to creatures."
Many spells affect a limited number of creatures
Typically your pet mouse would not count against the creature limit for entering a rope trick, being protected by a tiny hut, or coming with you on a teleport - although your DM may consider such an instance a case in which it would count as a creature. However, even the most gotcha-DMs are unlikely to say that your spell fails because of the fleas on the mouse, the lice in your hair, or the meal worms in your rations. Are all these things living creatures? Yes, in the biological sense, but in this case, they are not treated as independent creatures by the rules.
Background traits seldom have consistent mechanical effects
As Darth Pseudonym says, Urchin traits like food scraps are "roleplaying element[s] with no real mechanical benefit." The Urchin's pet mouse trait is similarly mostly for flavor, or perhaps an occasional situational bonus in the hands of a clever player, but not something you can rely on to provide an advantage in every encounter.
A rat, on the other hand, is large enough to have an attack, damage, AC, and hp - it has a stat block of its own and can serve as a familiar, designed to be used mechanically in many situations. But unless your mouse is large enough to get its own stat block, unless it is is big enough to be a threat and already have an attack in its own right, this DM would not allow it to gain an attack with the magic stone spell.
1Dampen elements: "When you or a creature within 30
feet of you...". Portent: "...by you or a creature that you can see...". Feather Fall: "...you or a creature within 60 feet of you...". Greater Invisibility: "You or a creature you touch...". Ottiluke's Freezing Sphere: "...you or a creature you give the globe to...". Phantom Steed: "...you or a creature you choose..."
2Note that an improvised weapon normally has a short range of 20 feet and a long range of 60 feet; at further than 20 feet a thrown object thus is at disadvantage to hit while the magic stone is not because it is a spell attack, not a weapon attack.
3Of cockatrice, the MM says (p.42), "These omnivores have a diet that consists of berries, nuts, flowers, and small animals such as
insects, mice, and frogs- things they can swallow whole." The size small cockatrice would not be swallowing whole something as a large as a rat.