The Oxford Guide to Style (2002) offers a usefully concise (but somewhat oddly punctuated) discussion of the colon at section 5.5:
The colon points forward: from a premise to a conclusion, from a cause to an effect, from an introduction to a main point; from a general statement to an example. It fulfills the same function as words such as namely, that is, as, for example, for instance, because, as follows, and therefore:
[Relevant example:] She had but one hobby: chocolate.
Various U.S. style guides adopt essentially the same view of the colon that Oxford does. All of them approve of the use of a colon to set off phrases such as the one in your example:
For three years, I ate hamburgers: my friend's favorite food.
Or, in Dan Bron's preferred formulation:
For three years, I ate my friend's favorite food: hamburgers.
Alternative punctuation options include an em dash:
For three years, I ate hamburgers—my friend's favorite food.
For three years, I ate hamburgers, my friend's favorite food.
and (arguably) parentheses:
For three years, I ate hamburgers (my friend's favorite food).