Update user's last login date
Here's some points. I commented directly in the code.

(function () {
    'use strict';

    //Why do you use `database` instead of `mongo` or `mongodb`.
    //It can become confusing later in the code when you are using `db`.
    //It will be hard to tell the diffence between database and db.
    var database = require('mongodb');
    var util = require('util');

    var constants = require('../constants');

    var dbhelper = {
        //`me` Will it be you all the time ?
        //You should think a more meaningful name such as `logedInUserId`
        updateLastSeenDate: function (me) {
            //Do you really want to connect and close the database everything you want to update a field ?
            //You should consider keeping the connection alive and share it across the whole process.
            database.connect(constants.variables.connection, function (err, db) {
                if (err) {
                    db.close();
                    throw err;
                }
                var bsonID = database.BSONPure.ObjectID(me);
                var collection = db.collection('users');

                //nesting calls to async function is affecting readability, testability and maintability.
                //You should consider using workflow library such as async
                collection.findOne({
                    _id: bsonID
                }, function (err, result) {
                    if (result) {
                        result.lastLoginDate = Date.now();

                        //Here's another nested call
                        collection.update({
                            _id: bsonID
                        }, result, function (err, u) {
                            //Using async you could handle errors in only one place.
                            if (err) {
                                db.close();
                                throw err;
                            }
                            util.log('Update complete: updateLastSeenDate');
                            db.close();
                        });
                    }
                });
            });
        }
    };

    module.exports = dbhelper;
}());


I would use async https://github.com/caolan/async#waterfall, change a few variable names and share the connection. Since node.js in single threaded you can open only one connection and share it to all your modules.