Carolle Ralph is stepping down from Moray Council after just six months representing Lossiemouth's Heldon & Laich ward for the SNP.
Police are investigating a series of letters which started after her election, and are appealing for information.
Ms Ralph said: "There comes a point where enough is enough.
"I stood for the council to work for the community, not to be a target for vicious bullying and a personal campaign of intimidation. My family life has been put under unacceptable stress as a direct result of this stalking and I'm not prepared to carry on in these circumstances.
"This is not a decision I have come to lightly. It is one of the hardest things I have ever had to do, given my commitment to this community which I love and which all my life has always been so important to me.
"I have received the kindest support from people in Lossiemouth and throughout the ward as well as from SNP colleagues. I would be grateful if anyone has any information that might help the police with their inquiries to contact them."
Grampian Police said: "We have received a complaint and are making inquiries."