In Ramayana, when and who named Sita mata as Vaidehi? What is the meaning of the name?
I found that Sita mata was called by the name "Vaidehi" only during the exile period. Is that right?
What is the meaning of Vaidehi, Janaki and Maithili?
Descendant of Vaideha
Princess of Vaideha kingdom
Descendant of Janaka
Descendant of Mithi
Princess of Mithila kingdom
Now question arises who was Vaideha or Janaka or Mithi? Or what is the meaning of these words?
After Nimi most of the king of his lineage came to be known as Janaka. As per Vishnu Purana, the Janaka who was Rama's father-in-law had name Síradhwaja.
As Nimi left no successor, the Munis, apprehensive of the consequences of the earth being without a ruler, agitated the body of the prince, and produced from it a prince who was called Janaka, from being born without a progenitor. In consequence of his father being without a body (videha), he was termed also Vaideha, 'the son of the bodiless;' and the further received the name of Mithi, from having been produced by agitation (mathana)
The son of Janaka was Udávasu; his son was Nandivarddhana; his son was Suketu; his son was Devaráta; his son was Vrihaduktha; his son was Mahávírya; his son was Satyadhriti; his son was Dhrisht́aketu; his son was Haryyaśwa; his son was Maru; his son was Pratibandhaka; his son was Kritaratha; his son was Krita; his son was Vibudha; his son was Mahádhriti; his son was Kritiráta; his son was Mahároman; his son was Suvarńaroman; his son was Hraswaroman; his son was Síradhwaja.
Síradhwaja ploughing the ground, to prepare it for a sacrifice which he instituted in order to obtain progeny, there sprang up in the furrow a damsel, who became his daughter Sítá. - Vishnu Purana - Part IV - Chapter V
I found that Sita mata was called as Vaidehi only during the exile period. Is it right?
Sita was called as Vaidehi by Rama himself when she was in Ayodhya:
And beholding the auspicious signs of pregnancy in his spouse Rama attained to excessive delight. Thereupon he said to the beautiful Sita resembling a celestial damsel "O Vaidehi signs of pregnancy are manifest in thee".
In Ramayana, when and Who named Sita mata as Vaidehi?
Your question is unclear to me. As it was the name given due to her family or kingdom, many people called her with this name. Even she is called as Vaidehi by Narada in the very first Sarga of Ramayana. And later on many times the name Vaidehi can be found in Valmiki Ramayana.
In ancient time a princess used to be named after the kingdom. For example, Panchali (Draupadi), the princess of Panchala and Gandhari, the princess of Gandhara.
Late Vedic literature such as Shatapatha Brahmana and Brihadaranyaka Upanishad mention a certain King Janaka as a great philosopher-king of Videha, renowned for his patronage of Vedic culture and philosophy and whose court was an intellectual center for Brahmin sages such as Yajnavalkya. Under his reign, Videha became a dominant political and cultural center of Bharatavarsha.
Videha had Mithila as its capital and was ruled by Rajarishi Janaka. Sita Devi was his adopted daughter. This is recounted in Valmiki Ramayana as well as Ramayana in other languages. Since Sita was the daughter of the ruler of videha, she was a princess of Videha.
The meaning of Vaidehi is “Of Videha; princess of Videha”. No one named Sita devi Vaidehi - it was automatic given her status.