Which is the right filiament type to print breakable children's toy parts such as small gears
deliberate/planned obsolescence is the term you look for
If you design parts that break after some time, you plan their obsolescence. That you do by a deliberate choice of material and working conditions. Designing a part that will break after a certain time can be done by choosing the correct stresses that will make your chosen material break.
In a gear that is meant to break at certain stress, one can weaken the teeth or the sprues, so that normal operation stresses will very likely break the safety margin and destroy the gear.
is it a material choice?
Any material is suitable to make a planned break, as long as the design is suitable. Performing a stress analysis of your part will tell you where to weaken it to enforce it will break - if the part was solid. As printed parts in FDM aren't solid, take the result with salt - it will tell where but not when it fails. Do the experiment for actual numbers.
is it a print setting thing?
Besides deliberately under-engineering some part of the gear, a usually perfectly fine gear would lose a lot of strength by deliberately reducing how massive it is: the stability of a print is affected by the form and amount of the infill just as much as the number of shells. Some random setting examples:

1-shelled, 1-bottom/top-layer, 5% infill piece is very likely so fragile you might not get it off the build plate
these parameters at 2-5-10% results in a somewhat durable piece.
2-5-20% is more than twice as strong as 2-5-10%.

To find the exact breaking point of a setup, one might need to toy with the parameters and experiment. It might be interesting to use no top- or bottom-layers and thus turn to create all the spokes of the gear in the shape of infill and outer shell. Also, some infills are better at withstanding forces than others - for example, Gyroid or Hex infill is rather stable on pressure while spaghetti is quite weak.
Other parameters also can change the infill stability: speeding up the print of the infill compared to the shell and using a thinner line considerably weakens the infill, thus reducing the needed load to break it. This is a somewhat easy parameter to tweak if you want to go for breaking the spokes (see below).
planned obsolescence and how to under-engineer safely
Sometimes, planned destruction is good for safety: a safety valve is supposed to break under overpressure to release the pressure in a safe way.
But planned obsolescence can also be a safety risk: If a toy breaks under normal use, it is a safety hazard for the broken off parts can be swallowed by children. Another factor to look at is where broken off parts end up in the machinery - they might jam other pieces that are not meant to self-destruct and destroy them.

Design the pieces to break in a safe way - the larger the chunks, the better you prevent them from going into places they should not.
Design the teeth to deform or melt rather than shearing off
Design the axles to sheer free by losing their keying
Design the spokes of the gears to break, separating gear rim from axle &amp; hub, either of which goes nowhere due to the other gears and the mounting
Encase the self-destruct gears in some sort of gearbox to prevent the pieces from going flying

Industrial machinery design usually goes the melting way: Let's take a hand mixer. It contains a gearset that has one drive gear connected to a second gear, so that both mixers spin opposite. Under normal use, these spin pretty fast, creating heat from the friction. In a good design, these two gears are made from metal or a high heat tolerant polymer. But if one plans for having them break, these gears are made from a material that will heat under the friction in such a way, that after a set time (around 5 minutes), the teeth will be sufficiently weakened to deform and grind away, destroying them in the process.
Preferred Material
I would actually deliberately under-design the gears for the expected loads and then go for a solid material printed in SLA or SLS from either a resin (which will break with pieces and bits going flying, so a gearbox is mandatory!) or a polyamide (nylon).  These parts would match the stress analysis fully.
If FDM is the only option, the material choice depends on the failure mode you opted for:

In case you opt for destruction from heat on the teeth or axle, a low melting material like PLA is perfectly fine, but make sure to engineer the chance of breaking teeth low. ABS can perform a little better but needs more heat (and thus more RPM) to self destruct.
In case of designing for a breaking failure of spokes or keying, PLA is an excellent choice, as it is sufficiently brittle.
PETG is a good compromise between ABS deformability and PLA's printing ease.


Footnotes
Gear DesignWhen designing your gears, keep in mind that gears are rather complicated. I actually advise to take a look on the gross oversimplification of This Old Tony because it allows you to see where you can make teeth break very easily by design!
planned obsolescence and consumer rightsWhile planned obsolescence can be an important safety factor, planning obsolescence in consumer products for sale to break them after a calculated time is unethical and can be a consumer rights violation. Remember, that legally demanded warranty and a right to repair exist in a lot of countries.
LEGO is Copyrighted, Patented and TrademarkedCopying Lego designs would be a Trademark Violation, Patent infringement and a copyright violation by using their designs. They protect them.