Classical Greek armies typically didn't have names, for the simple reason that they were not standing armies. When war broke out, a Greek city-state would call out its militia levy for a specific campaign; when the campaign was over, the men in the army would return to their regular civilian life. Not even an officer cadre remained of the disbanded army. A new conflict would require a completely new levy. Neither the total levy nor the smaller militia armies raised for longer campaigns received any names, numbers or insignia.

That said, armies that served together for a very long time might become known by a collective noun, usually based on the name of their commander. For example, the mercenaries of the Spartan general Brasidas fought with him for 3 years (424-422 BC), and even though Brasidas himself died in 422/1 BC, when his men formed part of the Spartan battle line in 418 BC they were still referred to as the Brasideioi. The mercenaries who fought for the Persian pretender Kyros the Younger became known as the Kyreioi (though they are now better known as the Ten Thousand).

In addition, some units were more permanently established, and these could have names. Here are some examples:

• According to Diodoros of Sicily, the Thebans had a special unit of 300 heavy infantry called the Charioteers and Chariot-Fighters. Diodoros is a very late source, and earlier sources do not mention this unit at all, but they may well have existed in some form. Their name would be a holdover from earlier times when its meaning would have been more literal.

• Similarly, the Spartans had a royal bodyguard of 300 heavy infantry, known as the Hippeis (horsemen). This unit is attested throughout the Classical period. It was considered the highest honour for a young Spartan to be selected for the Hippeis.

• The Spartan militia included a unit called the Skiritai. This was probably just the levy of a region known as Skiritis, on the border between Lakonia and Tegea. However, the Skiritai clearly served as a special unit of some kind. Several ancient sources credit them with a number of very particular tasks within the Spartan army, and we are told that the Spartans constantly used them for special missions while on campaign. They are definitely the most enigmatic named unit in Greek history.

• Around 421/0 BC, the people of Argos created the first standing unit known in Greece: a unit of 1000 hoplites maintained and trained at the expense of the state. They referred to this unit as the Thousand.

• Possibly in 378 BC, the Thebans created a picked unit of hoplites, which later tradition tells us was called the Sacred Band. It is not clear whether this name was really used in Classical times, because no Classical source ever calls them by this name, but it may well be the case that their permanent lodging and training on the city's akropolis led to them being associated with the gods worshipped there.

• In the early 360s BC, the newly formed Arkadian League raised a standing unit of an unprecendented size - 5000 men. They called this army the Eparitoi. The meaning of this word (unique to the Arkadian dialect of Greek) is lost.

OP mentions the perioikoi (latinised as perioeci), but this was not a special unit. The perioikoi were the freeborn non-citizen population of Lakonia, liable for military service when the Spartans called out the levy. At the time of the Persian Wars, they fought in their own formation, separate from full Spartan citizens, but by the time of the Peloponnesian War they seem to have fought together in an intermingled formation. They formed the majority of the Spartan army, rather than a named unit within it.