Basileia is defined in dictionaries as an ancient Greek word denoting ‘citadel’ or ‘royal palace’. However, Plato used both the Greek word ‘basileia’ as well as ‘metropolis’ in Critias 115c. Metropolis originally meant ‘mother city’ and in time became a generic term applicable to any seat of power. It is widely used in the New Testament with the meaning of ‘kingdom (of god)’.
Some commentators have used Basileia when referring to the entire capital city of Atlantis, while Jürgen Spanuth arbitrarily went further using Basileia as the name of the whole capital island, which he identifies as Heligoland. He also equates Basileia with Asgard, the home of the gods in Norse mythology. His views were echoed by others such as Walter Baucum.
When you combine that with Plato’s statement that names given to Solon were translated into their Greek equivalents and so we can safely conclude that Basileia, like Atlantis, was not an original placename.