My referrer logs tell me that people google "guerzhoy last name origin" sometimes.

The spelling of the name is a bit convoluted, but the pronunciation is simple: ger-JOY, with a hard "g" and the "J" pronounced like the "s" in "measure." The name is not as uncommon as it might seem: it was transliterated in my father's passport with the "u" when it was first written down in the Latin alphabet. "Gerzhoy" is the more common variant.

The simple answer to the question about the origin is that we don't really know.

My grandfather's Jewish and Yiddish-speaking family was from Pischanka. Pischanka was a shtetl in Ukraine, near the Moldovan border. The Yiddish spelling of the name is גערזשוי. In Yiddish, the "s" sound in "measure" (approximately) is denoted using a digraph, זש.

The name does not seem Slavic or Romanian, so the obvious candidate for the etymology is Yiddish.

But it's unclear that the name has a Yiddish etymology either. One theory is that it's a corruption of "Gershoi," which would come from adding the east-Slavic possessive suffix "oi" to the bibilcal name Gershon. (So basically "son of Gershon".)

A more fanciful theory I've seen is that it's a corruption of "Hirsch eug," Yiddish for "deer's eye." (Hirschauge does in fact exist as a surname.)

According to the extremely unreliable-seeming website, a boyar named Onuphrius Guerzhoy is mentioned in the church records of the town of Kursk in 1730. It's unclear what that means. A boyar named Onuphrius living in Kursk in 1730 would be very unlikely to have Jewish ancestry (Onuphrius is the name of a Christian saint, and the name itself is of Greek and/or Coptic rather than Hebrew origin; it's a plausible name for a Christian, but not for a Jew), and Jews were not required to adopt surnames until 1804. This seems to argue for a non-Yiddish origin. At the same time, because of the lack of an obvious Slavic etymology, we might imagine that Onuphrius or his family were converts.