His father Nimi left his body due to curse of Vashistha. The body of Nimi was churned and from the churned body he was born. Since he was born by Manthana (churning) he is called Mithi. Since Nimi was without body(Videha), so KushaDhwaja is called Vaideha. Son of KushaDhwaja is king Janaka father of Mother Sita.
His parents could not have any progeny, so Munis prepared mantra sanctified water for his mother. By mistake, his father Yuvanashva drank the water and conceived. So he was born by cutting his father's stomach.
All asked "Who will feed the child." So Indra came and put his finger in child's mouth and said " Maam Dhaasyati " (he will feed at me by finger). This is why he is called Maandhaataa.
Vyaasa's mother Satyavati was born when semen of a king Vasu was swallowed by a fish, the fish being an Apsara cursed to be a fish. Satyavati's body used to smell(Gandha) like fish(Matsya), so she was also called MatsyaGandhaa.
Later when Parashara Muni courted with her due to her beauty, he made her smell like lotus(Padma) and She was then called PadmaGandhaa.
4. Shuka Deva
Once Apsara Ghritaachi was wandering and unknowingly came in front of Vyaasa Deva. Due to fear she turned at once into a parrot bird (Shuki). But Vyaasa saw her beautiful female form, and later while preparing firewood his semen fell on firewood. When Vyaasa burnt the firewood then a beautiful boy like second Vyaasa came out of fire looking like fire. Since Vyaasa saw Ghritaachi in form of Shuki bird, he named his son "Shuka".
She is born of earth when king Janaka ploughed the ground.
Fire from Shiva's eye fell on sea. From that Jaalandhara was born. Since he was born of sea(JalanDhara) he is called Jaalandhara.
7. Astika Muni
To pursue for Sannyaasa his father JaratKaru was leaving his mother JaratKaru, so his mother JaratKaru said that they should have at least one progeny. So his father JaratKaru touched the navel of his mother JaratKaru and Muni Astika was conceived. (Yes both parents had same name) Mother JaratKaru is the great mother Manasa Devi , the goddess of snakes.
A king named Rabhya and his wife Rukmarekha performed a Yagya to get a progeny. A lot of Havishya was oblated, and from fire arose a beautiful girl of unparalleled beauty.
Since she came out like a whole garland, she is called "Ekavali"(Ek-one, Vali- garland). This girl Ekavali later on married to Ekaveera, son of Vishnu and Lakshmi in Horse-Mare form.
9. Muni Vritra
When Indra killed innocent Ascetic VishwaRoopa Trishira son of Tvashtaa, in fear of his throne, then Tvashtaa performed Abhichaara using mantra of AtharvaVeda. From the Yagya was born Vritra huge like mountain. Tvashtaa said - You are powerful and capable to save your father from troubles (Vrijrinaa), so you will be called as Vritra(Vri- troubles, Tra-to save from – one who save from troubles).
Vritra took great revenge from Indra and later with help of Vishnu Indra was able to kill him.
When Drupada king of Panchaal, did a Yagya to get progeny, Draupadi was born from the Yagya.