She said to him, "Why don’t you go today?"

She said to him why he don’t.

She asked him why he did not go that day.

She asked him not go that day.

None of these

The sentence is in direct speech, so it needs to be converted into indirect speech.

Rule: Change present tense to past tense, pronouns accordingly, and "today" to "that day".

Correct transformation:
"She said to him, 'Why don’t you go today?'" → "She asked him why he did not go that day."