The correct answer is “lay”.
The verb “lay” is used to describe the action of putting something or someone down in a particular position. In this case, the father is putting himself down in a particular position, namely, for a nap.
The verb “laid” is the past tense of “lay”. It would be used to describe the action of putting something or someone down in a particular position at a particular time in the past. For example, “My father laid down for a nap at 10:00 pm.”
The verb “lain” is the past participle of “lay”. It is used to describe the state of being in a particular position. For example, “My father has lain down for a nap.”
The verb “lie” is used to describe the action of resting or reclining in a horizontal position. It is not the correct answer in this case because the father is not simply resting or reclining, he is putting himself down in a particular position, namely, for a nap.