Many of the characters in The Matrix have meaningful names which relate to the role the character plays. For example:
- "Neo" is an anagram of "One" (for The One). Also, one of the many references to Neo as a Christ figure is his surname (Anderson), which means "son of man" -- one of the titles Jesus often used for Himself.
- Morpheus is the name of the Greek god of dreams -- Morpheus the character often speaks of the Matrix as a dream, as well as his dream that the Matrix would be destroyed.
- Persephone is the queen of the underworld in Greek mythology, and the character of the same name is the "queen" of the Matrix underworld controlled by her husband, the Merovingian.
Trinity's name is clearly related to the Christian Trinity (God is one but consists of three Persons -- the Father, Son, and Holy Spirit), but unlike the aforementioned characters it's less clear how her name relates to her role in the franchise. She doesn't seem to represent God since (a) Neo is much more clearly a God figure (specifically the Son of the Trinity) and (b) she dies but does not resurrect.
Perhaps Trinity's name is intended to point to the three main protagonists of the films (Neo, Morpheus, and Trinity herself). However, it doesn't make sense in-universe why Trinity would give herself that hacker name before such a trinity existed. Furthermore, while it's clear that Neo would represent the Son and Morpheus would probably represent the Father, it's not as obvious to see how Trinity would represent the Holy Spirit.
Have the Wachowskis explained what significance they intended for the character name "Trinity"? I am primarily interested in the out-of-universe reason for the character's name, though I am also interested in the in-universe reason why the character would give herself the hacker alias "Trinity" (especially if that reason sheds light on the out-of-universe significance of her character's name). If the Wachowskis have not spoken on the matter I am also interested in educated guesses as to the significance of the name.