per-: through, completely
sist: derived from Latin "sistere", meaning "stand"
So persist means:
Stand through (the whole time or process).
If someone wants to stand through the whole process of something, he/she must want to finish it completely. So it always means "to continue to do something, although it is difficult, or other people oppose it".
Evolution of usage
I will stand through (the whole time) until I succeed.
I will persist until I succeed.