Warren Buffett's mentor advocated true Value Investing — Quality and Quantity at a fair price — and not just plain cheap stocks.
Real Value Investing
The stock market does generally pay too much of a premium for good news. One probably does fare better with cheap, unpopular stocks.
But the crucial difference here is: cheap+decent stocks vs just plain cheap stocks. Plain cheap stocks are the domain of pure bargain hunting. Benjamin Graham advocated true Value Investing, i.e., Quality and Quantity at a fair price.
Warren Buffett
Buffett says that while he owes Charlie Munger and Phil Fisher a lot, it doesn't compare with what he owes Ben Graham. Buffett even named his son after Graham, and describes Graham's book - The Intelligent Investor - as "by far the best book about investing ever written".
Cigar Butt Stocks
NCAV stocks are simply the most well-known of Graham's strategies, and the source of the general misconception that Graham only recommended cheap stocks. But Graham actually recommended Defensive and Enterprising stocks before NCAV stocks and all were allowed higher Quantitative valuations and required greater Qualitative checks.
Graham did advocate paying more for quality. His only prerequisite was that there be the Margin of Safety between Price and Value, whether the Value be Qualitative or Quantitative.
Cigar butt stocks only refer to the simplistic NCAV stocks, and that too without the earnings and diversification requirements. Graham's more important stock recommendations are often ignored.
Buffett himself wrote an article called The Superinvestors of Graham-and-Doddsville [PDF] describing how Graham's principles are unquestionable and timeless.
But most of what Graham actually taught is no longer used today, and what he warned against is attributed to him instead. The Ben Graham formula is a classic example.
No Penny Stocks
The SEC defines Penny Stocks as any share trading below five dollars.
Penny Stocks as a category do not come within the purview of Graham's framework, as their only discerning feature is their low price. That presents no indication as to their underlying value.
A Penny Stock could be a valid Graham stock if it clears Graham's requirements for Intrinsic Value. But a Penny Stock would not be a valid Graham or Value stock simply by virtue of having a low price alone. Even a stock selling for 1¢ could be a bad investment if it represents a company that's hemorrhaging money.
All Graham Strategies
All of Graham's investment strategies are discussed in detail in How To Build A Complete Benjamin Graham Portfolio.
The last of the strategies is Special Situations. But Graham did not recommend Special Situations for the ordinary investor as it supposedly required a high degree of skill, experience and resources. But Buffett (being Buffett) has the skill, experience and resources; and almost all his operations do fall in Graham's Special Situations category.
Buffett has also explained how Graham was focused on refining methods that ordinary investors - without specialized access - could apply to achieve results similar to his own (Graham's).
All of Graham's students follow the same principles, they just apply them in their own ways. Buffett is simply the most visible of them because he's the wealthiest. But the difference between Graham and Buffett is only one of application, not principle.