Sue
Middle English: from Anglo-Norman French suer, based on Latin sequi ‘follow’. Early senses were very similar to those of the verb follow .
wiktionary
From Middle English seuen, sewen, siwen, borrowed from Anglo-Norman suer, siwer et al. and Old French sivre(“to follow after”) ( > French suivre), from Vulgar Latin *sequere(“to follow”), from Latin sequi. Cognate with Italian seguire and Spanish seguir. Doublet of segue. Related to suit.
etymonline
sue (v.)
c. 1200, "continue, persevere," from Anglo-French suer "follow after, continue," Old French suir, sivre "pursue, follow after, sue in court" (Modern French suivre), from Vulgar Latin *sequere "follow," from Latin sequi "follow" (from PIE root *sekw- (1) "to follow"). Sense of "start a lawsuit against" first recorded c. 1300, on notion of "following up" a matter in court. Sometimes short for ensue or pursue. Meaning "make entreaty, petition, plead" (usually with for) is from late 14c. Related: Sued; suing.
Sue
fem. proper name, a shortened or familiar form of Susan.