Kyria, the unisex Kyrie and the boy's name Kyrian are variants of the Greek girl's name Kyra, which currently ranks at #401 (in decline). They each mean "lord" in Greek but have a connection to ancient Persian and Egyptian in which they referred to the sun god Ra. The Greeks connected their word for lord to the Persian name for king, Kourosh (equivalent to Cyrus and possibly Cyril), which meant "throne." Several Persian kings were named Cyrus. In Egyptian, Ki-Ra meant "like Ra," or like the sun. There is no ancient connection to the word valkyrie, although if you were to name a daughter Valkyrie there is no sound reason to not nickname her Kyrie.
Kyria (KEER-ee-uh, and less often KY-ree-uh) is a variant of the popular girl's name Kyra. The spelling Kyria specifically means "noblewoman, respectable lady" and has been a Greek title of respect for women, much like "madam" or "miss." Kyrios was the male version of this title. It is a very rare name, with less than 40 girls given the name annually. The "Electa Kyria" was the second epistle of John in the Bible, and the woman to whom it is adressed is thought to be a noble woman from Syria who John cared for, I believe as a friend. She was considered humble and intelligent.
Kyrie (KEER-ee-ay, less often KUR-ee-ay or KY-ree) means "lord," and it has only ranked in the US top 1000 for two years now on the boy's side, rising quickly from #868 in 2012 to #590 in 2013. It does not currently rank for girls, but there are over a hundred born each year. During mass "kyrie eleison" are the only Greek words spoken, and they mean "Lord have mercy."
The boy's name Kyrian (KEER-ee-an, less often KY-ree-an) is the male variant of these names, and a more modern-boy look to the title "kyrios." There is a character named Kyrian that was created by author Sherrilyn Kenyon. As mentioned above, Cyrus (Ciro, too) is another Greek boys name meaning "lord," from the same origin.
Kira is also taken to be a variant of Kyra, although Kyra is sometimes said "KY-rah," not "KEER-uh." Kira in Russian is taken from the masculine Kir, and means "mistress, ruler." Kira is also a Japanese name.
There are towns named Kyra in both Cyrus, Greece and Zabaykalsky, Russia.