Step 1: Understand the question.
The question asks about a scholar who held that the mantras of the Veda have no meaning. This is a famous old debate in Indian thought on the meaning of Vedic hymns.
Step 2: Recall the source of the debate.
In his Nirukta, Yaska names the scholar Kautsa. Kautsa argued that the Vedic mantras are 'anarthaka', that is, without meaning. Yaska then replies to him and defends the meaning of the hymns.
Step 3: Check option 1, Yaska.
Yaska is the author of the Nirukta, the work on word origins. He answered the charge and held that the hymns do have meaning. So he is the defender, not the person who argued they are meaningless. This option is wrong.
Step 4: Check option 2, Panini.
Panini wrote the Ashtadhyayi, the great Sanskrit grammar. He worked on the rules of language and did not argue that the hymns are meaningless. This option is wrong.
Step 5: Check option 3, Patanjali.
Patanjali wrote the Mahabhashya, a commentary on Panini. He is also known as the author of the Yoga Sutras by tradition. He did not make this claim. This option is wrong.
Step 6: Check option 4, Kautsa.
Kautsa is the one who held that the Vedic hymns are meaningless, and Yaska records and rejects this view. So option 4 is correct.
Final Answer:
Kautsa claimed that Vedic mantras have no meaning, and Yaska answered him. This is option 4.
\[ \boxed{\text{Kautsa (option 4)}} \]