What kind of relay do I need?
If you have 3V to drive the relay, then you can possibly use a Solid State Relay. It seems to be just barely enough to turn it on.

However I feel that if you are asking this kind of question, you shouldn't be modifying the AC wiring in your home just yet. It could possibly kill you and others either by electrical shock or fire.

Please be careful and research a bit more, possibly for other kinds of controllable lights which you can also modify and hack.