What is the Purpose of Own Antyesthi (Cremation) before taking Sanyasa?
It is because the rites are performed when the body dies. Similarly when one takes up Sannyasa, it is symbolic that His body attachment is now dead. That is one has cut off all ties with His physical body. Therefore the cremation rites are performed.
Paramahansa Yogananda describes this in Autobiography of a Yogi, Chapter 24:
The bibidisa or elaborate initiation into swamiship includes a fire ceremony, during which symbolical funeral rites are performed. The physical body of the disciple is represented as dead, cremated in the flame of wisdom. The newly-made swami is then given a chant, such as: "This atma is Brahma" or "Thou art That" or "I am He."
However some saints do not perform such initiation. For example, Paramahansa Yogananda did not do this ceremony, because Sri Yukteshwarji (His Guru) decided that is was not needed.
Paramahansa Yogananda describes this in Autobiography of a Yogi, Chapter 24:
Sri Yukteswar, however, with his love of simplicity, dispensed with all formal rites and merely asked me to select a new name.
Note: This DOES NOT implies that these rituals are NOT NECESSARY AT ALL. This is just example to show that not all saints need to follow these rituals. User moonstar2001 brought the issue in comments below, thus I am clarifying it here.
I think that also would answer your other question:
On which Basis(Spiritual Qualities) does Guru gives a new name to Sanyasi or Shishya(Disciple)?
There is a general guideline for this based on Dashanami tradition for monks of Shankara's order of swami's. Paramhansa Yogananda describes this in Autobiography of a Yogi, Chapter 24:
In addition to his new name, usually ending in ananda, the swami takes a title which indicates his formal connection with one of the ten subdivisions of the Swami Order. These dasanamis or ten agnomens include the Giri (mountain), to which Sri Yukteswar, and hence myself, belong. Among the other branches are the Sagar (sea), Bharati (land), Aranya (forest), Puri (tract), Tirtha (place of pilgrimage), and Saraswati (wisdom of nature).
But I think you mean to ask for: based on these guidelines, how exactly the name is chosen? For example, how did Sri Yuktheswarji choose 'Yogananda' as the title? How did Sri Ramakrishna choose 'Vivekananda' as the title? How did 'Chandrasekharendra Saraswati' get his title?
There is no rule for this. The Guru simply follows His Intuition or inner Guidance or Dharma. Whatever you call it. Usually Guru gives the name to His disciples, but sometimes the Guru also asks the disciple to choose it Himself! For example, Paramhansa Yogananda was asked to choose His own spiritual Title:
Paramahansa Yogananda describes this in Autobiography of a Yogi, Chapter 24:
"I will give you the privilege of choosing it yourself," he said, smiling.
"Yogananda," I replied, after a moment's thought. The name literally means "Bliss (ananda) through divine union (yoga)."
"Be it so. Forsaking your family name of Mukunda Lal Ghosh, henceforth you shall be called Yogananda of the Giri branch of the Swami Order."
Is there any Co-relation of above two:
The only correlation is that only the One who has give up all ties to body and physical differences can take up the name of Swami. Thus the two are actually correlated in the sense one is a pre-requisite for the other.
Paramhansa Yogananda describes the qualities of Sannyasis in Autobiography of a Yogi Chapter 24:
The ideal of selfless service to all mankind, and of renunciation of personal ties and ambitions, leads the majority of swamis to engage actively in humanitarian and educational work in India, or occasionally in foreign lands.
Imbuing his waking and sleeping consciousness with the thought, "I am He," he roams contentedly, in the world but not of it. Thus only may he justify his title of swami one who seeks to achieve union with the Swa or Self.
It is needless to add that not all formally titled swamis are equally successful in reaching their high goal.
How about Swami Vivekananda?
Swami Vivekananda, after Sri Ramakrishna Paramahamsa's Mahasamadhi, took up Sannyaas with eight other disciples, and thus took the title of Sri Swami Vivekananda which means the Bliss of Viveka (or Discrimination).[ Reference: Wikipedia (Not sure how accurate this is though)]