Step 1: Meaning of the phrase.
The idiom “the apple of one’s eye” means someone or something that is greatly loved, valued, or cherished.
Step 2: Analyze the options.
- The most tender part of an eye: Literal meaning, not correct.
- A tearful eye: Incorrect.
- Very dear: Correct, as it reflects the figurative meaning.
- (A costly treatment of an eye: Incorrect.
Step 3: Conclusion.
Thus, the phrase “the apple of one’s eye” means “very dear.”
Final Answer:
\[
\boxed{\text{very dear}}
\]