A divorce is often like a death, and if your neighbor is like most people in her situation she's grieving the death of her marriage -- even if she doesn't realize it. And sometimes the best thing we can do when someone is grieving is simply to let them know we care, and to keep quiet and listen to them if they choose to talk.
In other words, treat your neighbor the way you'd want to be treated in her situation. If you were in her shoes, you'd probably feel hurt (or even angry) if people ignored you or acted as if nothing had happened. And yet this often happens when someone experiences a deep personal tragedy such as a divorce -- often people are embarrassed and don't know what to say.
Tell her you know she's going through a rough time and you want to be her friend. Even if she seems uninterested at first, keep reaching out to her (without being a pest). Realize that the time may come when she will reach out to you. The Bible says, "As God's chosen people ... clothe yourselves with compassion, kindness, humility, gentleness and patience" (Colossians 3:12).
Most of all, pray for her, and as you have opportunity encourage her to turn to Christ. God loves her, and even when we pass through some of life's hardest places, God's peace and hope can be ours. It's true for her -- and it's true for you.