Derives from the Latin mater meaning mother.
At first brush the Matrix is equivalent to Hinduism's Maya, the world of illusion. The Matrix is unreal in the sense that it is 'only in the mind' of humans. The reality on the planet during the 23rd century is that homo sapiens are being grown by machines to harvest energy to power themselves. The machines are keeping and tending a human farm. The 'life' that humans perceive is a production by an outside force to keep them unconscious.
Thomas "Neil" Andersen.
He is of course the vaunted Messiah that has been prophesied to save the world (humanity). The name Thomas is Jewish meaning 'twin.' This name is significant given that Andersen has two personalities. As one of the Agents had explicitly told Thomas during the interrogation he has a legitimate life as a software engineer for a big company and simultaneously the underground life of a computer hacker. We can also say that Thomas has a twin personality in that on the one hand he was living an illusion wherein the only reality for him was the Matrix, and on the other hand he was destined to break out of the Matrix and to break the stranglehold of the Matrix, of Maya, on the whole of humanity.
Furthermore, and more importantly, the name Thomas alludes to the idiom 'doubting Thomas.' Andersen does not believe he is the Savior. He is actually initially nonplused at the suggestion. Only when he commits to saving Morpheus from the Agents does he gradually realize his mission, his purpose, his destiny, his vocation in life. As in real life unless one is in the thick of things, until one has become aware of the risks of commitment yet nevertheless musters the courage to commit one will not know nor taste fulfillment (or defeat). Two roads diverged says Robert Frost; to stand forever at the fork is to forfeit the right to one's destiny.
'Neil' is an Irish name meaning 'champion.' Andersen opts to go by this name rather than by his given name.
