Do you mean the city in Libya? If so, the answer is definitely not because it simply pre-dates Cyrus' life.

The birth of Cyrus is typically placed around 590 BCE. His wars of conquests began in 553 BCE, and he died in 530 before he ever had a chance to campaign against Egypt, let alone the rest of North Africa. Persian influence only reached Libya after Cyrus' son, Cambyses, conquered Egypt in 525. Even then, the Persians did not intervene in Libya directly for another decade. At first, their control of Cyrene and its neighbors was purely an extension of the tribute Cyrene had paid to the independent Egyptian pharaohs before the Persian conquest. Only around 515 BCE, when the Persian satrap of Egypt intervened in a Cyrenean civil war, did the Persians and Cyreneans interact directly.

According to the Histories of Herodotus, the city of Cyrene was founded by Greek colonists from the island of Thera who fled their island following a catastrophic drought. The famed Oracle of Delphi advised the Therans to settle in the relatively fertile region of Cyrenaica on the north African coast. We don't know where the name "Cyrenaica" comes from, but it was already the established name for that part of the Libyan coast in the 7th Century BCE. Presumably, it was derived from a name in a local Libyan (ie Berber/Amazigh) language.

After a few false starts on the coast, the colonists took the advice of the local Libyan people and moved further inland, where they established the city Cyrene around 631 BCE. Their coastal settlement, Apollonia, was maintained by a small remaining population as a port.

This colony was initially ruled by King Battus I, and by the time Cyrus the Great was a young man, Cyrene was on its third king, Battus II, another delegation went to Delphi to ask for advice on improving their city. The Oracle advised them to recruit more colonists from the rest of Greece and settle new territory, and that anybody who didn't join them would eventually regret it. Because of the Oracle's prestige, that led to a rush on Libya and the rapid foundation or expansion of two more Greek colonies ruled by Cyrene.

In the following decade before Cyrus the Great even embarked on his conquests, the emerging kingdom of Cyrenaica underwent a civil war that led to some rebels fleeing and founding the city of Barca, which was ultimately defeated and assimilated into the rest of Cyrenaica. A fifth city, Euesperides was founded by more settlers from Cyrene itself shortly before the Persian conquest of Egypt.

So not only was Cyrene not named for Cyrus, but it had become the nexus of its own small kingdom long before Cyrus' people ever made contact.