Is there a way to make an LED blink using a passive circuit?
You need a wheel. A bicycle wheel should work, and the flashing light makes sense on a bike. Some other wheel may work, depending on what do you have at hand. Maybe a fan would work.

Buy a magnet and reed contact like these. Connect battery, LED, resistor and the reed contact in place of the switch. Now just approach the magnet to the reed contact, it should trigger, lighting the LED.







simulate this circuit &ndash; Schematic created using CircuitLab

Attach the magnet to the wheel and place the reed contact nearby. As long as the wheel is spinning (by hand, or by motor if it is a fan, or bike is moving) the LED should flash when the magnet is passing near the reed.

As @Wossname commented, a pendulum would also work instead of a wheel. You can use an old pendulum clock without any electric components.

To satisfy all requirements of the question "in depth", you would need a small electric engine rotating the wheel, probably with reductor. Or a steam engine if you really dislike the electricity, up to you. However the mechanical complexity may not outweight the lack of one another transistor.

BTW the reed, the LED and the resistor all have nice, long pins, easy to wire, and the child may be happy to play just holding the magnet in his hand.