Story time, kids! Once upon a time (July ’08) in a corner of the web far, far away, a search engine named Cuil challenged Google. While claiming to search more pages of the net than anyone else, it was found to be slow — and came under fire from critics — and quickly fell back into insignificance.

Since then, Cuil (pronounced “cool”) has tried to stand out by including supplementary search results, such as including timelines and geographical “maplines.” Now, Cuil would like to further beef up its results by including related results from social networking sites such as Facebook and Twitter. According to Seval Oz Ozveren, the engine’s VP of finance and business development, it’s something that hasn’t been done before.

“We are trying to leverage the information found on users social networks to enhance search results,” he told the Technology Review. “This is similar to what Amazon or eBay already does: ‘People who bought this book, also liked this one’ … nobody has actually done it to date.”

While social networking sites are currently searchable through internal means and by way of specifically tailored apps, Cuil’s method would put it out there along with whatever you’re searching for. This has some pull in two big ways: socially driven advertising is a hot new market and, especially when it comes to Twitter, breaking news and viral fads are appearing first on social networks.

The engine plans to include the new functionality in August. Whether or not it will help Cuil recapture the magic and go toe-to-toe with Google — something that Bing, which hasn’t been crushed, is arguably doing — will have to be seen.

So we know you Google. We’ve asked if you Bing. (That just sounds dirty.) Is anyone out there staying Cuil?