Alexandros

Very popular on its native turf, foreign-sounding here. 

Alexios

Anatolios

Andreas is a beautiful name, with the patina of an Old Master painting, one that could make a dashing namesake for an Uncle Andrew. 
 In Europe, Andreas is currently in the Top 20 in Norway and is also popular in Denmark, Sweden, Germany, and Austria.
Some other international variations of Andrew you might want to consider — Andre, Andrei, Andres, Andrea, and Anders.
 Notable namesakes include photographer Andreas Feininger. One not-so-pleasant association: the San Andreas fault line in California.

Andreas

Though this name is rooted in Greek mythology, it just entered the US Top 1000 for the first time in 2013, but as one of the year's fastest-rising names.

 In Greek mythology, Ares was the blood-thirsty god of war and son to Zeus and Hera. Names from ancient myth and legend are generally on the rise, as is evidenced by Gwen Stefani's choice of Apollo for her third son.

Ares

Atlas is one of those names that was previously thought too powerful for a baby boy, who would have to be strong enough to carry the world on his shoulders. Now Atlas has joined the pantheon of Greek and Roman god and goddess names in the realm of possibility, along with Mars, Zeus and Apollo.

 Atlas was one of the fast-rising names on the list in recent years in the USA, jumping from oblivion in 2012 into the Top 500 in 2015, and climbing several hundred places higher since then. Anne Heche was one of the first to make this audacious choice, but several celebrity parents have followed suit.

 As well as the mythological connection, Atlas is a cool geographical word name – perfect for the child of parents who love traveling or who hail from different parts of the globe.

Atlas

Achaikos

Agamemnon

Agathon

Alcibiades

Anaxagoras

Andronikos

Apollon

Aristides

Ariston

Arkadios

Artaxerxes

Bion

Cadmos

Chariton

Charon

Chrysanthos

Cimon

Coriolanus

Damon is a name with a strong, pleasing aura (much like the persona of Matt D.) and extremely positive ancient associations. From the classical myth, Damon and Pythias have become symbols of true friendship, as Damon risked his life to save his friend from execution. And Damon of Athens was the fifth century philosopher who taught both Pericles and Socrates.
 Other interesting associations: Damon Runyon wrote Guys and Dolls, Damon is one of the funny Wayans brothers, Damon Dash is a high-profile music exec, and Damon Salvatore is a key figure in The Vampire Diaries.

Damon

Demetrios

Demonax

Drakon

Herakles

Isidoros

One of many M names -- including Murray, Melvin, Morton, Milton, and Marvin -- given to first-generation Jewish boys to replace the old-fashioned Moses. Now we'd pick Moses over any of them. 
 Myron was the name of an important 5th century BC sculptor and also of several saints. In the US, it was a Top 200 name in the 1930s.

Myron

Mentor

Midas

Nikomedes

Nikandros

Nikanor

Niketas

Nikias

Nikodemos

Nikomachos

Nikon

Philon

Prometheus

Despite the reputation of the sagacious ancient Greek lawmaker, this name hasn't moved to the modern world. 

Solon

Simonides

Theodoros

Theron is best known these days as the surname of actress Charlize, but it does have a long history as a first name, ranking in the Top 500 in the early part of the 20th century and only dropping out of the Top 1000 in the early 1990s.  Theon is a similar name made familiar by the popular series Game of Thrones: Might it and Theron rise in tandem?

Theron

The name of the Greek mythological hero famous for slaying the Minotaur is also heard in Chaucer and Shakespeare. While Theseus might have sounded pretentious for an American baby a decade ago, today with so many babies getting ancient mythogical names, it just might work. 

Theseus

Thanatos

Timoleon

Ragnarok's list "Ancient Greek Boys' Names" of 50 great name ideas: Alexandros - Timoleon!

Ancient Greek Boys' Names

Ancient Greek names, including from history and mythology.