Eulogios of Edessa was the first Eulogios I could find.
Moreover, on Febrouary 13, Eulogios, the archbishop of Alexandria
, is honored:http://www.missionstclare.com/english/people/feb13o.html
Hence, even nowadays, a Greek child born on February 13 could be given the name Eulogios.
I am native Bulgarian. (Bulgaria is the northern neighbour of Greece.) Definitely, there is a Bulgarian name Evlogi. If you check the Bulgarian name calendar in this site, you would find Evlogi on February 13. The Greek name calendar in this site misses that day but I think it is dedicated to Eulogios, anyway.