This variant is called Teleporting Kings / Teleporting Kings 960. The rules are pretty simple. Almost all of the rules are the same as in chess or Chess960/Fischer Random (whichever you choose to play) except each king has the ability to teleport once per game.

Teleportation allows the king to move to any square on the board with some restrictions.

Restrictions:

• The king cannot teleport to a square that is occupied by any piece by either player. The king cannot use teleportation to capture an opponent's piece.
• The king cannot teleport to any square that is being attacked by the enemy. The king cannot teleport into check.
• The king cannot teleport while in check. The king cannot teleport out of check.
• The king cannot teleport to a square that he could move to without teleporting. Any king move that can be made without teleporting is considered a normal king move.

Things to note:

• Teleportation is considered a king move. If the king teleports, castling is no longer legal.
• Unlike castling, the king's ability to teleport is not affected by any of the king's previous moves (besides previously teleporting). If the king has already teleported, teleportation is no longer legal.
• The king's position on the board does not affect his ability to teleport. There does not have to be a clear path to a square that the king teleports to. Even if the king is completely surrounded by the player's own pieces, teleporting is still legal.
• Note if playing Teleporting Kings 960: If the position is such that king-move-only castling is legal, all of the requirements to king-move-only castle are met, and the player chooses to move his king to the square the king would land on if he did king-move-only castle, then the move is considered a castle. In this situation, the king preserves his ability to teleport later in the game.

Let me know what you guys think.