Hey Robness,
Thanks for your question and welcome to the community.
The best base for your tile is to use tile backerboard in a 1/2" thickness. You'll need to clear out the existing edges of the hole to allow yourself to set a rectangular piece slightly smaller than the hole (about a 1/4").
To allow the backerboard to stay, use tile backerboard tape (shown below) and overlap it on the new backerboard piece as well as the existing wall.
Be aware that you'll most likely need to take out those adjoining tiles that aren't affected just so the tape can securely be affixed to the base. Once this is done, you can apply a thin-set mortar rated for vertical surfaces.
During each and every step, always ensure that the the backerboard, the tape, the mortar, and the tile are completely level to the rest of the wall so it looks like no repair was ever done in that location.
The mortar will go over gap where the tape covers, as well as adhering the tile to the wall. Once it is dried and the tiles are successfully installed, then you can seal everything and the repair is complete.
At each step, work patiently and slowly to ensure everything is set in place properly.
Let us know if you have any further questions,
Joseph