The line of dialogue that is punctuated correctly is:
"Our lease says no pets," his mother replied.
The other examples have punctuation issues:
- "Why don't you start a dog walking business," his sister replied. (should end with a question mark instead)
- I want a pet, James said. (should have a comma before "James" or a period at the end)
- "But I really want one", he whined. (the comma should be inside the quotation marks)