Doorbell for someone with hearing issues
I know you are keen on using your Arduino.  But simpler solutions abound.  For instance, if the door bell already works at a lower then main voltage it might be easier to determine the feasibility of connecting a low voltage lamp to the door bell.   

However, I can see a situation that an Arduino can fit very nicely into:

Events: 1) The door bell rings. 2) The Arduino senses this and starts turning on and off a room light at about 5 second intervals. 3) This can last until someone notices and resets the Arduino.  4) If no one is home to reset the Arduino, the Arduino stops turning on and off the room light after a minute passes (12 flashes).

There are 2 tricky bits that may need explaining.  The first is how to sense the door bell.  If mechanical you could use a micro switch:



That way you don't need to worry about isolating the Arduino from what is likely a strong electrically noisy solenoid.  The other tricky bit is controlling a room light.  I would suggest using a COTS remote controllable light switch (I just did a search and there are loads of them in every price range) and hack the (battery powered) remote control into the Arduino.