CZroe is on point. If the brightness button turns off the light, it's a 001 and isn't worth the trouble. You can also post a photo of the cart connector side of the board to verify the model.
It's 100% a power issue, if not something else in addition. I believe the power line to the LCD is one of the most outer wires in the ribbon.
Possibilities:
Blown regulator, which can happen because of a maligned ribbon cable or bad battery/charger. This means the mainboard is toast.
Damaged ribbon. This means the LCD module is toast. You'll have to take everything apart and play with the cable while the power is on to test.
Bad battery. Easy fix
Bad game cartridge (believe it or not). Easy fix, change the game.
Water/humidity damage. It's only worth saving the LCD in the case, and only if it's a 101. There are little square stickers inside the unit that change color with water damage.
Also, bravo to Phynn for putting together a list of names.