Thursday Caliban is mentioned in The End as the husband of Miranda Caliban and the father of Friday Caliban. Thursday is a V.F.D. member who is assumed to have sided with the volunteers. His family arrived on the island with the Baudelaire parents, and helped with the functioning of the island until Ishmael arrived, lobbying for the island to be made more safe from the treachery in the world and the Medusoid Mycelium.
From what is implied in The End by Ishmael, Thursday left the island shortly after the Baudelaire parents left. It can be assumed that he and Miranda disagreed as to whether they should return to the world after what happened in V.F.D.. Because of this disagreement, Miranda told her daughter Friday, after she was born, that her father died during the storm that washed her up on the island, and was eaten by a manatee. This makes absolutely no sense, as manatees are herbivores, but as Friday has known nothing but the island, she would not know that this isn't true.
Not too long before Kit Snicket washes up on the island, she said that she had met with Thursday, and even spoke with him about the state of the island before the departure of the Baudelaire parents. Because of the joyful way in which he described the island to Kit, it is unsure whether or not he is aware that he has a daughter, or that his daughter believes that he is dead.
In the commonplace book A Series of Unfortunate Events, it is noted by Beatrice Baudelaire that a postcard from Kit addressed to Olivia Caliban washed up on the coastal shelf of the island. From this, it can be determined that Olivia Caliban is Thursday Caliban's sister.