I never understood how progeny of Rishis were blessed/allowed to start new Gotra system. The progenitors already had their Gotras. So what was the need for sub Gotra system?
Example, Sandilya Rishi was allowed to start gotra of his own. He as son of Rishi Asita but there is no gotra in the name of Rishi Asita. Rishi Kashyapa had his own gotra and he was father of Asita and grand father of Sandilya.
So, what parameters were seen in their progeny to allow them to start new Gotra system?
I had a same question few years back and I was not able to find the reason. Very recently I did some ferreting on Google and found an interesting article on speakingtree on Gotra system. It is as accurate as it can be. For an example, I belong to Sāṇḍilya Gotra- Sāṇḍilya rishi, the son of Devala rishi and the grandson of Kashyapa rishi who did penance to please Goddess Saraswati at Sarda Peeth in today's Kashmir Valley. He was blessed by Goddess Saraswati that his work will remain for ages to come and thus happened. Sāṇḍilya rishi's work Sāṇḍilya Upanishad became integral part of Chandogya Upanishad which is widely read today also. Thus his progeny and student who studied under him adopted this name as Gotra.
Gotra in the Hindu tradition can be translated as lineage or the clan to which a person belongs. Every Hindu is assigned his or her gotra during birth. As long as girls remain in their father's house, they belong to their father's gotra. Upon marriage, they enter their husband's gotra. Thus, gotra is usually the lineage referred to in case of the male leader of the family that is successively passed on through the male heirs.
In the Hindu tradition, there are four varnas namely Brahmin (the priestly class), Kshatriya (the ruling class), Vaishya (the trading class) and Shudra (the working class). Among these four classes of people, gotra applies only to the first three and not the Shudras.
In the Brahminical tradition, the gotras have their origin from the seven rishis also called as the Sapta rishis namely Gautama, Bharadwaja, Vishvamitra, Jamadagni, Vashista, Kashyapa and Atri. Later, sage Agasthya was also added to this list making the number eight. These rishis have their origins directly from the creator namely Brahma and all their descendants are addressed therefore as Brahmins or those that belong to the lineage of Brahma. Later, this kind of system was adopted by Kshatriyas and Vaishyas too.
In some tests, it is noted that the entire set of disciples who learnt from a particular rishi in the gurukul or school established by him came to be called by the rishi's name referred to in their gotra and thus all of them came under a gotra eventually continuing throughout their lineage.
While talking of gotra, we must distinguish it from kula. While the term kula refers to the people following some common set of cultural rituals and often found worshipping the same kula devata (deity of the kula), this term does not refer to the clan or lineage.
Gotras are also noticed in the families of Shudras. However, the gotra system in this case is typically different from that of the other three classes. For instance, a weaver might say he belongs to the Markandeya gotra. Markandeya was a rishi who had sixty sons. In such cases, the marriages are held within the Markandeya, but not within the same family name. This means that under Markandeya, there will be several families with different family names.
In Hindu marriages, it is a common practice to enquire about the gotra of the family in which a marriage alliance is sought. Marriages are never permitted within the same gotra meaning that the bride and groom of sharing the same gotra are related to each other as brother and sister. Therefore, marriages are performed between the families of different gotras.
Though the gotra system in Hinduism has its origin in the Vedic classification of lineages initiated by the seven sages, eventually, it got expanded to be applied to different classes. In several cases, it referred to the guru or the preceptor under whom the early people got their education. The present day gotra system in Hinduism is linked to all lineage systems that have different origins other than those of the Vedic origin too.
