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As a native speaker, I know instinctively that number 1 is correct but I'm finding it difficult to summarize this for my students in an understandable way.
The simple agreement rule* is that the number of the verb should correspond to the number of the grammatical subject. The Cambridge Grammar of the English Language (p499) states the rule as follows:
In simple agreement, the verb agrees with a subject with the form of an NP (noun phrase) whose person-number classification derives from its head noun.
The head noun in this case is food, so sentence 1 (My favorite food is hamburgers) is the right choice.
Garner, in American English Usage (p778) in the section False Attraction to Predicate Noun notes the temptation use the plural verb in such contexts:
Occasionally a writer incorrectly looks to the predicate rather than to the subject for the noun that will govern the verb. The "correct" way of phrasing the sentence is often awkward, so the writer is well advised to find another way of stating the idea.
One example of "false attraction" listed by Garner is:
You can use live or artificial bait to catch these fish. My favourite are top-water plugs, plastic jigs and live green backs or shrimp.
* There are many occasions when the simple rule stated above is overridden. For example, Ten dollars is a lot of money for a hamburger or My family eat fish every Friday.
Food here means 'type of food' so it takes a singular verb.
In British English, hamburgers refers only to the patties, so (1) would be the most natural. In American English hamburger can mean the minced (ground) meat from which burgers are made, so using it as a non-count noun as in (3) would be possible.
