Brilliant pebbles were a concept created for then-President Ronald Reagan
's not-so-ill-fated Strategic Defense Initiative
, or Star wars
to its critics). They were (in technical terms) coast-phase ballistic missile warhead bus
Deciphered, this means that they were a system or set of systems designed to intercept and destroy ICBM warheads between the point in their flight profile where they stopped boosting and the point where the warhead bus began to dispense MIRVs. In point of fact, they were to be able to attack dispensed MIRVs as well, but were designed to hit the unitary bus package for reasons of efficiency and increased Pk (Probability of Kill).
The name reflected their design philosophy. Essentially, the near orbitals would be sown with either multitudes of robotic KKVs, or with dispensers designed to place them rapidly and accurately near the flight path of an oncoming missile. Using onboard guidance, they were to maneuver themselves in front of the target in order to force a collision, which would (at the speeds the warhead was travelling) destroy the warhead or at least (likely) damage it to the point where it couldn't dispense MIRVs.
The problems with this approach were and are legion. First of all, how does one get the KKVs into position? Although there are limited arcs of flight for inbound ICBMs into the U.S. from the former USSR, orbital dynamics means that it wouldn't be possible to sow the entire area with permanent or even semi-permanent clouds of KKVs. They would need to be dispensed, which meant likely being fired in a missile from an orbiting base, which caused a whole host of its own problems, not least of which was that such an installation is expressly forbidden by treaty.
Finally, it was never convincingly demonstrated that the KKVs could be made to work, especially at the price points that would be required to field 'clouds' of them. Orbital maneuvering isn't trivial, and requires decent computers; finding the target warhead far enough out for the KKV's limited specific impulse to generate an intercept would require large, fragile and expensive sensors, and so on.
More expensive, unitary hit-to-kill interceptors appeared to have a much higher degree of feasibility and likelihood of success. Flight tests for an interceptor KKV were carried out at Lockheed(?) in the final days of the SDI heyday; they looked like nothing so much as a coffee can hovering in the air over a net, maneuvering around and maintaining altitude with sudden bursts from its jets. As far as I know, however, this was never married to a booster for a live-fire test.
A more limited version was seen in prototype, named FLAGE, designed to go after slower and easier IRBM (theater ballistic missile) targets.