Dear Sri Prasad,
Mother of this child seems to be interested in getting her child initiated as a Brahmachari.
We should go to the purpose of wearing Yagnopaveetham.
Let us see what the Yagnopaveetha Sangalpam says:
After declaring the Samvatsarm, Ayanam, Ruthu,Masam, Paksham, Thithi, Day, Nakshatram, Yogam, Karanam etc,
it says the purpose of wearing the Yagnopaveetham in the following words:
Sri Bhagavathagnaya Sriman Narayana preethyartham, Chrowtha smartha vidhi vihitha nithya karma anushtana yogyadha sidhyartham Brahma thejaha Abhiviruthiyartham Yagnopaveetha dharanam karishye.
This is in simple Sanskrit, I hope will be understood by all. Thus the purpose of wearing Yagnopaveetham is to bestow
the right to follow and perform Nithya Karma Anushtanam as enunciated by shruthi (vedas).
If some one is earnestly interested in putting his/her son in Vedic studies, let he/she follow the rituals as per the custom. She can adopt her Gothra or the Gothra of the Acharya.
There are instances where the name of the Acharya or his Gothra was adopted by the students.
My best wishes to the Brahmachari,
Regards,
Brahmanyan,
Bangalore.
This is one of the best answers with lots of clarity, Lokah samastha Sukhino Bavanthu