Byzantine archaeology (6):
http://www.worldheritagesite.org/categories/category27.html

Göreme
Daphni, Hosios Loukas and Nea Moni of Chios
Mystras
Nessebar
Thessalonika
Um er-Rasas

I would also add: Istanbul, Ravenna, Abu Mena, Pecs (?), ancient villages of Northern Syria, Troodos, Aquiliea, Bethany, Porec, Patmos, Kosovo, Moldavian churches (?), Mt. Athos, Ohrid, St. Cathrine, Stari Ras and Sopocani, Studenica, Palermo/Monreale and Kiev (?).

My selection:

Göreme - for underground architecture and frescoes
Mystras - for a complete Byzantine city.
Istanbul - for Hagia Sophia and the Chora church.
Thessalonika - maybe for the churches
Ravenna - frescoes
Abu Mena - a well preserved Byzantine pilgrimage site.
Pecs - for early Byzantine burials.
Moldavian churches - for the exterior frescoes.
Mt. Athos - for a long standing monastic tradition.
St. Catherine - same.
Ohrid - For a well preserved Byzantine city.

I always feel that those "Byzantines" get a bit over-represented!! A "classic(al)" example of over-representation of a culture/period vis a vis others around the World!!
Of the 6 "official" ones under this category only
Goreme is YES (and that is as much for its other aspects as for its Churches

With further (almost certain) selected coverage under other categories for - Istanbul, Ravenna, Mt Athos and, St Catherine they are still doing very well!

Browsing through the Wiki-pages for Byzantine Art and Byzantine Architecture, I'd say the following have been the most influential:
- Early Period (- 842 AD): Hagia Sophia in Istanbul, Rotunda in Thessaloníki, St. Catherine's Monastery in Egypt and the Basilica of Ravenna
- Middle or High Period (- 1204): Monasteries of Daphni, Hosios Loukas and Nea Moni of Chios & Monreale, Cefalu and Palermo (!)
- Late Period (-1453): Mystras ("when traditional mosaic-work gradually gave way to detailed cycles of narrative frescoes")
- Post-Byzantine: should cover Byzantine influence into Bulgaria, Serbia, Romania and, most importantly, to Russia,

So from the Original list I'd pick these 3:
- Daphni, Hosios Loukas and Nea Moni of Chios - representing the High Period of Byzantine Art and architecture
- Mystras- representing the Late Period of Byzantine Art and architecture
- Thessalonika - representing the Early Period of Byzantine Art and architecture

Istanbul, Ravenna and St. Catherine's Monastery would certainly qualify for this category too, but they've already been covered at others and will get in via those.

We might need to look at the Post-Byzantine / Eastern European ones separately.

Why is "Byzantine" Art/Architecture so important among World-wide styles/periods that it needs to be represented across its Early, Middle, Late and "Post" periods? This would seem to be simple "Euro-centricity".

The Byzantine Empire also was very influential in the (Near) Middle East. And the mix with Islamic architecture shows f.e. at Palermo/Monreale and Jerusalem's Dome of the Rock.

We have a Connection of 52 sites related to it, so a selection of (in total) 6 of those does not seem excessive to me.

I think I'm with Solivagant on this one. I don't know if representatives of various stages of the style are needed, in the same way I wouldn't promote various versions of Gothic architecture need representation.
There will likely be a fair few Byzantine sites included for other reasons (Jerusalem, Istanbul, Ravenna, Mt Athos perhaps Kiev? will be included under other categories) and they will cover many of the stylistic aspects.

So out of the original 6 I would only go for Goreme.

To put it differently we haven't looked at early, middle and late Maya for example. There are also countless cultures and styles we are not going to represent at all.

My selection:

Ravenna
Goreme
Moldavian Churches
Troodos churches
Istanbul
St Catherine

I'll only comment on the original list of 6 sites, as the others seem to have been discussed somewhere else already. Out of this category, I would only select Göreme (iconic rock-cut dwellings) and Mystras (preserved Byzantine town).

I would vote only for Göreme National Park

If you are looking at 52 sites, then we have many connection into A-list . (like Athos, Meteora, Ravenna) and Many are yet to come out of that list (Istanbul,Venice, Rome)

I Second with Assif regarding Maya. Mayan culture has impact on meso -America like no other culture. It can also divided into various stages. We are also looking at a regionally balanced list. Lets take best sites to represent the architecture style. Göreme National Park is clearly shoo-in.
( Alongwith Istanbul, which is not here)

This one is still missing a summary.

Summary:

Göreme
Daphni, Hosios Loukas and Nea Moni of Chios
Mystras
Nessebar
Thessalonika
Um er-Rasas

I'll bring these over to the main page http://www.worldheritagesite.org/allbyrank.php as well.