If something is persistent, it just won't stop. If you're persistent in your suggestions for a trip to Mexico, maybe your parents will eventually give in and take you there.
Persistent comes from the Latin verb persistere which means "to continue with strength." A persistent salesman refuses to give up. A persistent ringing sound is one that repeats and repeats. Persistent rumors keep circulating, as does a persistent virus. Often something persistent is annoying. The related verb is persist, which means to continue to exist, or to go without stopping.