When my daughter was eight one of her friends at the time suddenly turned into a bully and gave her no peace. There was nothing physical, just meanness and rudeness.
I told my daughter one thing bullies hate is cheerfulness and politeness. We practiced her responses to things the bully would say.
It worked wonders and drove the bully crazy enough that within two or three days she started avoiding my daughter.
Try not to take over and deal with this yourself. Direct parental involvement lets the bully know that they have succeeded in their aim: to cause distress.
My daughter was verbally bullied when she was eleven and again at fifteen. We taught her to be nice to the bullies, cherish the friendships she had and to grow and nurture new friendships. The bullies never knew they upset her and soon became figures to be pitied by other kids.
Invite the bully to your home and give them a really nice time with games, tea or a sleepover. Hopefully they'll want to come back and you might even find out what is making them bully your child.
Get on the internet and look for sites on bullying. Childline is quite a good one.
I think when the children are young it is easier to give them solutions, by suggesting responses to bullying and practicing what to say. It is harder and more complicated when they're older.
My younger brother was bullied almost all through his elementary school years. My Mum put him in a private institute for a year and between that, and my other brothers taking him along on activities, he has found things that he likes and is good at. Now he knows how to stick up for himself and though he still struggles in school, it's easier for him because the bullies don't want to bother him.