You need to build up your $set
object programmatically:
var setModifier = { $set: {} };
setModifier.$set['directions.' + index + '.name'] = area.value;
Items.update(Session.get('selectedItem'), setModifier);
Update
If your JavaScript environment supports computed property names (e.g. node.js 4+), you can do this in one step:
Items.update(Session.get('selectedItem'), { $set: {
['directions.' + index + '.name']: area.value
}});