As far as I can tell, the only reason to stay at the Byzantion is to visit the site at Mystra. It's well located for that -- a very short drive to the parking for the main gate. It's also right by the handful of restaurants and souvenir shops that make up the current village. The downside to the location was that my room was right across the street from a restaurant where lots of people were eating outside so it wasn't particularly quiet during the evening and early night.
There's a small parking lot. Based on my stay, it's just barely adequate at night. (Much thanks to those who helped me with parking in an unfamiliar car.) The rooms are comfortable if not particularly generous in size. Oddly, the room while non-smoking had an ashtray, supposedly for use on the balcony. The breakfast seemed fairly standard.
The one real problem that I had was the wifi -- I could only get it working in my room for about 10 minutes late in the evening. It did always work in the lobby. Based on my interaction with the staff, I gather that they're aware that some people have trouble with the wifi but are puzzled.