First, of course, it could have reference to the kindness of her own words. In this sense, it would mean that her speech (her "tongue") is kind, that the "teaching of kindness" is manifested in what she says. If this is its meaning, then it is sympathetic with the first part, because wisdom is always kind. By emphasizing this aspect of wisdom, the writer identifies what is most evident about her speech - its kindness. In her wisdom, she speaks the appropriate word, the word that her listener needs to hear, in the way the listener needs to hear it. Scripture tells us that kindness and gentleness characterize all that the godly do in relation to others. We are to instruct in kindness and even to rebuke in kindness when someone sins. The kindness of the excellent wife is a great gift to those who know her. Her words are like silver.
Second, it can mean that she actually "teaches kindness" to others, that is, that she gives instruction, not just example. Some translations adopt this meaning when they translate it as "faithful instruction" rather than "teaching of kindness." If this is its meaning, then it means that, being wise in her own speech, she teaches others the same practice. She instructs her children in kindness, of course, and others as well (you would expect). How does she teach them? It does not say, except that she speaks to them.
A woman of wisdom, who speaks with wisdom, will find many opportunities to teach others. People will share their concerns, their frustrations, their anger over things that happen to them, and she will have the chance to instruct them in how to respond appropriately. We see this a lot in women's meetings with friends. Women, like men, often share their hurts and their anger with one another. What does a woman of wisdom say to a young wife who is angry at something her husband has done? What do you say to someone who comes to your lunch get-together seething with anger?
Too often, women believe the best option is to share her own complaints or "build up her sister" by agreeing with her. Meetings can become "gripe sessions" where each woman leaves convinced that she is right. Instead, there needs to be a wise woman in the group who can tone it down and teach kindness.
The woman of wisdom not only manifests kindness in her words but also in her actions and her teaching of others. She calms the angry waters and builds up the heart and spirit of everyone with whom she speaks. She sends people back to their lives with something extra for those with whom they deal, a touch of kindness.
She opens her mouth in wisdom and the teaching of kindness is on her tongue.