http://bingo.cdyn.com/
http://www.ballybingo.co.uk/index1.html
http://ballybingo.nl/Engelse_index.html
http://red-rollover-lit.blogspot.com/
http://archive.ipdb.org/russjensen/inside.htm

Take a lot of pic's before and while you are taking something apart. Your biggest regret will be, "oh, I'll remember how to put that back together". No you won't.

I would clean the light socket until it is shiny where the bulb sits/locks in. If that didn't help I would replace the bulb. If that didn't work I would clean the contacts on the Relay that powers that bulb.
Along the way look for loose solder joints, Loose bulb sockets and loose women. Just kidding about the women!
Hope it works out, bright lights are all about the right voltage getting to the bulbs contact points.

The 3 P's:

Patience
Parts
Perseverance

Make sure that the Jones plugs are clean and tight.

Awww, come on guys, it's so simple maybe you need a refresher course. It's all ball bearings nowadays. Now you prepare that Fetzer valve with some 3-in-1 oil and some gauze pads, and I'm gonna need 'bout ten quarts of anti-freeze, preferably Prestone. No, no make that Quaker State.

Loved that movie.

Copper will get a layer of corrosion that will block voltage. Happens at work all the time. Tests no voltage scratch the copper and wow 12,000 volts appear. Good luck.

I’d recommend starting with the bingo repair videos. I’ve watched both videos three times. If you know as little as I did (some would say, still do), it is money well spent. It helps if you compile your own table of contents as you watch the videos. For example, you might record “Search Disc - volume 3, about the 5.23 minute mark.” It will save you time when you need to go back later for a refresher.

Alan Marriage video:
ebay.com link: Bally EM Bingo Pinball Machine Service DVD Video Volumes 1 2 NEW SEALED

Richard Gerlitz video:
ebay.com link: Bally Bingo Pinball Repair Course 11 Total Videos Get your game running

The Gerlitz video is better of the two, but it’s useful to see two guys who both know their stuff explain things their own way.

Phil Hooper’s web site has the most detailed explanation of bingo circuitry you will find. Start with the “How They Work” section. Make sure you have a bottle of scotch so you can follow along in full fidelity:
https://bingo.cdyn.com/techno/howtheywork/

Ask questions here on Pinside, you’ll get plenty of help.

Check out the Bally Bingo’s in Britain forum. There are lots of knowledgeable and helpful collectors there, many of them European:
http://ballybingo.fff.yuku.com/forums/2/Maintanance-and-Repair/Maintanance-and-Repair

Are the numbers that are dim on card 4 and 6 different numbers? Barrel O' Fun works a bit differently than the non-six card games in that the lamps are tied together on cards 1-3 and 4-6.

Don't forget to post some pics. I can never get enough love seeing different games.