STEP 1:
Select the object then press "f". This switches you into UVface select mode.

STEP 2:
Now press "w" to bring up the specials menu.

STEP 3:
Select "Light", then press "f" again. Now try running the game engine and see if your object is lit now.

Another way to do this is to select your object and press "f" to enter UVface select mode.
Now press F9 to enter Editing mode. Activate the "Light" button under the "Texture face" Tab. Then hit "copy DrawMode" at the bottom of that same Tab.