How did Garuda got his name as Garuda (गरुड)?
Garuda’s mother, Vinata (विनता) mother of the birds, was tricked into becoming the slave of her sister Kadru (कद्रू) , mother of the nagas (serpents). So to free her mother from being a slave, Lord Garuda agreed to bring amrit (अमृत, nectar) for nagas. During his pursuit to bring amrit (अमृत), he stopped to meet his father (Rishi Kashyapa).
There his father narrates him a story of elephant and tortoise who use to live in a nearby pond. Garuda’s father tells the Garuda that this elephant and tortoise were two brothers in a previous birth, and had the names Vibhaavasu (विभावसु) and Suprateeka (सुप्रतिक). They fought for family property and when they remain undecided about the same. They cursed each other, and as a result one brother is born as an elephant and other as tortoise. But the fight, which began in the previous life, still continues!
The two brothers now reborn as animals, so to liberate them from their current birth, sage Kashyapa ask Garuda to eat them.
Garuda comes upon the two animals, and he picked elephant in his one claw and tortoise in another claw and he flews towards Meru Mountain. There he saw a big tree and tried to sit on a branch so that he can eat both of them. But due the weight of Garuda the branch fall off from the tree. So Garuda quickly catches the branch in his peak.
During that time Garuda carried elephant in one claw and tortoise in other claw and also holding the fallen off branch on which Valakhilya Rishis (वालखिल्य) were hanging. Due to this capacity of him to hold & carry heavy weight, he is called as Garuda ( गरुडम् आदाय उड्डीन इति “गरुड:” )
गुरुं भारं समासाद्योड्डीन एष विहङ्गमः।
गरुडस्तु खगश्रेष्ठस्तस्मात्पन्नगभोजनः।। – Mahabharata Adi Parva 30.7
King of Birds and the serpent killer who wanders in the sky and flown with heavy weight, therefore he is called Garuda