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This is what I want:
A system for enchanting pet rocks, miniature non-ambulatory golems, that can be placed in unassuming places around your home or campsite to sense intruders and make some sort of alarm.
What could I use in the game to make that happen?
XGE popped into my mind as it reminded me of the associated picture. Wizards are given a level 3 spell called Tiny Servant (XGE, 168):
You touch one Tiny, nonmagical object that isn’t attached to another object or a surface and isn’t being carried by another creature. The target animates and sprouts little arms and legs, becoming a creature under your control until the spell ends or the creature drops to 0 hit points.
As a bonus action, you can mentally command the creature if it is within 120 feet of you. (If you control multiple creatures with this spell, you can command any or all of them at the same time, issuing the same command to each one.) You decide what action the creature will take and where it will move during its next turn, or you can issue a simple, general command, such as to fetch a key, stand watch, or stack some books. If you issue no commands, the servant does nothing other than defend itself against hostile creatures. Once given an order, the servant continues to follow that order until its task is complete.
Using that, upcast as a 4th-level spell, you can have up to 3 pet rocks and command them "bash this plate if someone approaches", Since they have a movement of 120 feet and requires no concentration, I think that might hit your target.
The Wizard who specializes in the School of Transmutation can, at sixth level, create a Transmuter's Stone that stores transmutation magic. You can have one active stone at a time.(PHB, p. 119)
The rules do not explicitly allow you to put a Magic Mouth or Alarm spell on it, but since you can imbue it with infravision out to 60' (explicitly provided for in that class ability), and Darkvision is a second level spell effect, you can probably convince your DM that a spell of similar or lesser power level may be imbued into this stone (Magic Mouth is a second level spell). Alarm being a first level spell may an even easier sell to your DM.
The DMG (p. 129) allows for enchanting some items on (Crafting Magic Items). Since you only need to be 3d level to cast that spell, you may be able to argue for a Common or Uncommon item classification. However, since it's a sixth level class ability, a more conservative DM may call for it to be a Rare item. The cost and time to create such an item of the agreed quality is spelled out explicitly in the DMG - your DM will have his input as well (it's a Downtime Activity).
The table on page 285 of the DMG would support this being an Uncommon item with a spell power greater than 1 and less than 3 in the case of Magic Mouth, and a Common item in the case of the first level Alarm spell.
If you go the "Craft a Magic Item" route, don't bother with the Transmuter's Stone. Just work with your DM on the cost and time to make the Alarm rocks per the table on page 285, and the details on page 129.
To put an Alarm spell on one rock should take you 4 game days and 100 gp.
This answer was based primarily on the core rules (PHB, MM, DMG) but if you have / use Xanathar's Guide to Everything at your table, you could instead use Tiny Servant as explained in VictorB's most excellent answer.
Seems like you're looking for an "Alarm" rock, acting much like the spell of the same name. A good skeleton to build upon would be a Common Figurine of Power (dmg 169).
Mix in the Crafting Magical items guidance from DMG 128 and the Dungeon Master's workshop follow on (DMG 285). The crafter should be a magic user of at least level 3, that has the ability to cast Alarm. Expected duration is 4 days for 100g total. Depending on how variable the Alarm effect would be, increase the cost to make.
Usable once per day, require the user to be attuned. Through that attunment, the user would be alerted via the mental option of Alarm.
