If you persists in telling lies to me I shall sue you ...... slander.

If you persist in telling lies to me, I shall sue you for slander.
In this context, "persist" means to continue firmly in an action or course of action despite difficulty or opposition. "Slander" refers to the action of making false and damaging statements about someone. Therefore, "for" is the correct preposition to use after "sue" to indicate the reason for taking legal action, which is the act of slander.