The verb persevere comes from the Latin root persevereus which means "continue steadfastly, persist" and that word comes from two others, per- for "very" and severus for "strict." So, today, the connotation of the word is to persist in a methodical way despite obstacles or distractions. You can see the relationship between persevere and perseverate, which means "to persistently repeat something," and that word is a medical term used to describe stuttering.