關 is a compound of 𢇇, which represents threads or ropes binding the beam of a portal, 門 being a gate, it means "barrier, frontier pass, close, relation".
東 is traditionally thought to represent a sun rising behind Fusō, a mulberry tree in the far east where the sun originates according to Chinese mythology. It means East.
Together, in the context of Japanese history and the real-life region of Kantō, they represent the region east of the Sakone pass, the main tollgate between two historic economic centres of Japan, Edo and Kyoto. We can also interpret the name in the context of the Pokémon world as east of the Victory Road gate at Mt. Silver.
Johto = 城都
城 is a compound of 成, representing an axe with the handle to the right and the blade to the left, with the little half-square representing the border of a city, and 土, a lump of dirt representing earth, together they represent a castle or urbanity in general.
都 is a compound of 者, a sugarcane with its stems and a mouth, meaning to boil (original version of 煮, later borrowed to become a nominalising suffix), and 邑, a person kneeling outside of a city, together meaning a metropolis.
Together they represent Ecruteak, the traditional capital of Johto, and its iconic Tin Tower and Brass Tower.
Hoenn = 豐緣
豐 is a pot of plants 𠁳 on top of another pot 豆, representing fertility and abundance.¨
緣 is a compound of 彖, a pig with a prominent snout meaning "hog", and 糸, silk threads, together meaning the hem of a cloth, its edge. It semantically shifted to mean edges in general and eventually destiny, serendipity.
Together, it has been stated by Masuda to represent the serendipitous prosperity of humans and Pokémon, but we could also see it as representing the ideologies of Team Magma and Aqua, a destined land of abundant land or water.