Before the Day of Lavos
Mother Brain was a central computer that linked together all of the domes and factories in the year 1999 AD. After Lavos emerged and destroyed the world, Mother Brain took control of all the robots on the planet. Since they were not alive, Lavos did not—and perhaps, could not—consume their energy, leaving them to thrive. The factories of Geno Dome and the Derelict Factory worked night and day, constructing more machines.
Three Centuries Later
By the year 2300 AD, Mother Brain had built a massive army of robots. Crono, Marle, and Lucca came through time to stop it. They destroyed its central unit, thus freeing the surviving humans from the threat it posed.
Following the destruction of Lavos back in the year 1999 AD, the timeline shifted to one where the adventurers never came to destroy it. When Belthasar arrived in the future, as he was always destined to, he created a gigantic supercomputer named FATE from designs of it and Robo, and hard-wired it into his research facility: Chronopolis.
The three Displays surrounding Mother Brain heal it. Each Display has a single Hit Point, so they are easy to destroy. However, once they are destroyed, Mother Brain starts using more powerful techs against the group. It cannot physically attack nor counterattack, nor is it resistant or immune to any element. Using Techs that damage multiple targets will destroy the Displays and provoke Mother Brain's lethal attacks. Instead, use Lucca's Hypnowave to inflict the Status Effect Sleep on these computer monitors. This prevents them from healing Mother Brain and leaves it is unable to attack the party. From this point, focus on single-target Double Techs such as Fire Sword or Blaze Kick to destroy Mother Brain. After it is destroyed, the battle ends.
A Mother Brain is to be an adaptation of motherboard, the central circuit of all computers, without which the entire machine would cease to function. As its duties include monitoring and programming the other robots of this time period, its absence might render disparity in the memory banks of machines, causing them to malfunction, lack a unifying force, or simply obtain individuality of thought. Additionally, the word mother assists this idea, as mothers govern, love, teach, and instruct their children. Brain, in the name, may allude to its nucleus like functions. Brains are the commander center and engine of the bodies of higher animals, which is identical, again, to the role it plays. Lastly, Mother Brain denotes a sort of Swarm intelligence