Concept:
The definite article “the” is used when a particular person, thing, place, or event is specifically identified in the sentence.
Step 1: Identify whether the noun refers to something general or specific.
The noun used here is “lunch”.
The sentence does not refer to any lunch in general. Instead, it refers to a particular lunch that was offered by the speaker.
\[
\text{lunch we offered}
\]
Thus, the noun is specific.
Step 2: Apply the rule of the definite article.
Whenever a noun is made specific by additional information, the definite article “the” is generally used.
The phrase “we offered” clearly identifies which lunch is being discussed.
Step 3: Construct the complete sentence using the appropriate article.
The sentence becomes:
\[
\text{They really enjoyed the lunch we offered.}
\]
This sentence is grammatically correct and conveys a definite meaning.
Therefore,
\[
\boxed{\text{the}}
\]