I don't know that why River Ganga is also called "Bhagirathi River"?
This has to do with how Ganga, a river which flowed in the Svarga, was brought to Martyaloka. The legend as told in the Ramayana goes like this,
When king Sagara chose to perform the Ashwamedha yagna, his royal agents lost track of the sacrificial horse. Sagara ordered his sixty thousand sons by Sumati to track down the horse. The proud and mercurial prince raged across Bharat, burning down forests and uprooting life and property to find the horse. They finally arrived at a quiet spot where the Sage Kapila was sitting in meditation. Beside him was tied the white horse. The enraged prince condemned Kapila as a thief and attacked him. When the sage opened his eyes, his immense power turned the princes into ashes. The Kosala kings of successive generations could not do this while managing their duties as kings. As a result, the sins of the thousand princes multiplied in their destructive energy, and began resulting in natural disasters. The kingdom began to lose its peace and prosperity, and by the time Bhagiratha ascended the throne, he found it impossible to govern.
Bhagiratha turned over the kingdom to his trusted ministers and set off to the Himalayas to perform an arduous tapasya in the extreme climate. For one thousand years, he performed an excruciatingly harsh penance to please Lord Brahma. At the end of the thousand years, Brahma was pleased and asked him for his wish. Bhagiratha asked Brahma to bring the river Ganges down to earth so that he may perform the ceremony for his ancestors.
Brahma asked Bhagiratha to propitiate Lord Shiva, for only He is capable to break Ganga's landfall. Ganga has a strong flow and it would have been impossible for anyone to contain the destructive impact of this event except Shiva.
Bhagiratha performed a tapasya for Lord Shiva, living only on air. The compassionate Shiva appeared only after a year's penance, and told Bhagiratha he should not have to perform tapasya to accomplish a noble goal such as this. He assured Bhagiratha that he would make Ganga fall on his matted locks (dreadlocks).
In honour of the person who was responsible for bringing Ganga to the Earth, Ganga is also called BhAgirathi (meaning: of-Bhagiratha).
She flowed in a destructive manner and destroyed Jhahnu rishi's ashram. This angered Jhuhnu and took Ganga as ApOzana. Bhagiratha again prayed for Jhuhnu to let out Ganga and Jhuhnu let her out. Hence, Ganga is also known as Jahnavi. Ganga then flowed over the remains of the 60,000 ancestors of Bhagiratha and brought them moksha.
As already answered in Why and when did the Ganga become a holy river?, Ganga was brought by Bhagiratha on Earth by penance, so She is called Bhaagirathi.
Devi Puraana, Book 9, Chapter 6
गङ्गे यास्यसि पश्चात्त्वमंशेन विश्वपावनी ॥ ४९ ॥
भारतं भारतीशापात्पापदाहाय पापिनाम् ।
भगीरथस्य तपसा तेन नीता सुकल्पिते ॥ ५० ॥
नाम्ना भागीरथी पूता भविष्यसि महीतले ।
मदंशस्य समुद्रस्य जाया जायेर्ममाज्ञया ॥ ५१ ॥
मत्कलांशस्य भूपस्य शन्तनोश्च सुरेश्वरि ।
Vishnu tells Gangaa-
Then, O Gangaa! You will go in some parts of yours, , to the BharataBhumi, due to curse of Bhaarati(Saraswati), to become the purificator of worlds and to burn the sins of sinners.
O well armed Gangaa! by Bhagiratha's penance You will be brought down, , and will be known as Bhaagirathi, and will be most sanctifying on Earth.
By my orders, You will become wife of Samudra, who is I in part incarnation. . And also, You will be wife of king Shantanu, who is I manifest in part incarnation.
The story goes like this. A long time ago, there was a king named Sagara. He has sixty thousand sons. One day, King Sagara decided to perform rituals for the betterment of his kingdom. There was a horse which was an integral part of the ritual and it was stolen by Devaraj Indra in jealousy.
King Sagara sent his 60 thousand sons all over the earth in order to find the horse. After searching everywhere, they found the horse next to a meditating sage Kapila. The horse was tied next to him by Indra. Believing that Kapila Muni (sage) has stolen the horse, they disturbed his Tapasya (meditation). Angered by this act of his sons, he cursed them and the sixty thousand sons were burnt to death.
The souls of the sons of Sagara wandered as ghosts since their final rites had not been performed. For the moksha of the sons of Sagar, Anshuman (nephew of those 60,000 sons) prayed to Brahma failed. His son Dilip did the same but failed.
Bhagirath was a Suryavanshi king who was one of the descendants of Sagara, son of Dilip, learned of this fate. Saddened by their fate, he wondered if there was a chance that they might live again, be reborn or attain moksha.
He went to Kapila muni and ask him if there was any way by which his ancestors can attain moksha or reborn. Kapila Muni told him that they could be reborn only if their ashes were poured in the Ganga. For this Ganga would have to descend from heaven to earth. He then vowed that he would bring her down to Earth.
Bhagirathi did Tapasya and pleaded Brahma that Ganga comes down to Earth. Brahma agreed and he ordered her to go down to Earth so that the souls of Bhagirath’s ancestors could be freed. She agreed, but she said that the force of her fall would end up destroying the Earth. Bhagirath then prayed to Lord Shiva and requested him to hold her on his head.
She falls on the Shiva’s head and he contained the force of the Ganga in his locks and let her out in small streams. Bhagirath was able to immerse the ashes of his ancestors in the river and they were freed and reborn.
Because of Bhagiratha’s efforts, Ganga descended to Earth and hence the river is known as Bhagirathi.
If you want to know more about River Ganga, read these two articles: