The supplies should cost about $10 or less. You will need:
1x pumpkin (of course!)
1x Blue LED
Stiff wire (coat hanger, florist's wire, etc.) to maintain the growth's shape
Electrical wire (insulated or enameled) - at least a couple feet. Thinner wire will be easier to hide.
Battery holder (4x "AA" or "AAA" (etc.), or 9V battery snap) and batteries
Resistor (value depends on battery holder)
Hot-melt glue and gluegun
Step 1: Choose Your Battery Holder and Resistor
A resistor's "stopping power" or resistance is measured in Ohms; the higher this value, the more strongly it resists the flow of current. A typical blue LED can handle 20-30mA of current. For either type of battery holder, a resistor value in the range of 250 to 560 will work fine (you can use higher values to make the LED dimmer). If you are using a 4-cell holder, you can go as low as 100 ohms.
1/4 Watt is the most common size of resistor you will find, and will be fine for this project. You can use larger wattages (1/2W, etc.) if you have one lying around, but I wouldn't recommend going smaller. Since it's hard to inscribe a numeric value on the round package (and harder still to read!), resistors at this size are normally marked with colored bands to indicate the resistance. If you are faced with a package of assorted resistors without numeric values, look up "resistor color code" for help in identifying the correct one.
Step 2: Building the Lantern Appendage
Take about 6 inches of stiff wire (a piece of coat hanger or floral wire work nicely) and bend it into the desired shape of the lantern appendage. This will form the center of the lighted appendage and help it maintain its shape. In a real anglerfish it protrudes from the forehead and dangles tantalizingly over its gaping maw, but arrange it how you like. Squirt a dab of hot glue at the forward end of the stiff wire, then quickly (before the glue hardens) touch the LED to the glue so that it sticks, facing forward. If your stiffener is bare metal, make sure that any exposed wires/metal from the LED do not come in contact with it, in order to prevent a short-circuit.
Lead the wires back toward the rear of the stiffener, tacking them to what will be the underside with a bit of hot glue at intervals. Leave the last couple inches of stiffener free of glue so that it can be easily poked through the jack o'lantern's forehead.
Now that everything is held in place, goop all but the bottom couple inches of the stiffener with hot glue, wires and all. Starting at the LED, run beads of glue over the stiffener toward its base, working your way around until it is completely covered, and as lumpy and organic as desired. You'll want to add a nice thick blob of hot glue around and in front of the LED itself, this will diffuse the light and help it spread down the metal stiffener. Don't worry too much about the wires being visible right now; in the dark, the bright glow from the end will help conceal them.
Step 3: Carve the Pumpkin
If possible, save the plug(s) you carve out so they can be placed back in at night to slow this process. Alternatively, rubbing a light coat of olive oil over the thin parts should help keep the moisture in.
Step 4: Wiring and Finishing
Wire the battery holder and resistor to the LED lantern's wires as shown. Remember to be sure the anode (longer lead; positive) you identified earlier is wired to the + side of the battery. If you have wired everything and the LED does not light, try reversing the wires to the LED.