There are in fact records of multiple persons with that name, and in fact there are several of them in the Bible. For starters, "Jesus" is not the name that the man would have been known by in his lifetime. "Jesus" is an Anglicization of Iēsūs, a Latinization of the Greek "Ἰησοῦς" (Iesous), which is itself the Greek form of the name Hebrew/Aramaic "Yēshūăʿ". Jesus, the son of Joseph, would have been known to those around him as "Yeshua ben Yoseph". To add another layer to this, Yeshu'a (יֵשׁוּעַ) is a form of the name "Y'hōshūă" (יְהוֹשֻׁעַ). Yehoshua in turn gets Anglicized as "Joshua", which leads eventually to the culmination here, namely that "Jesus" and "Joshua" are, through a convoluted path, the same name. The name itself derives from "God [Jah] is salvation".

There are in fact several persons then who bear this name in the Bible. The second-most famous of course would be Joshua, of "The Book of Joshua", and one of the great heroes of Jewish history, leading them into the promised land and defeating the Canaanites. In this light, giving your child the name of "Yeshua" or "Yehoshua" is not unlike the old practice in early America where the Founding Fathers provided inspiration for names of children like Washington Irving, or perhaps something like the common Armenian name of Haik, which harks back to their mythical founding figure.

Many other Joshuas (Yeshua/Yehoshua) existed in those times, such as the Biblical figure of the High Priest in the Book of Zechariah, as well as more tangible evidence such as tomb stones found on various graves in the region from the period. Ilan and Hünefeld provide a number more examples from various literature of the period such as Rabbinical writings and Josephus. Some of you will no doubt remember the minor news item from a decade ago around the 'Talpiot tomb' in Jerusalem which was revealed to have the names of Jesus, Mary, and Joseph present on ossuaries (it had been discovered years prior, but that year Discovery Channel did a nice sensationalist pseudo-history piece on it). Needless to say, controversy surrounds it, but at the very least a strong argument against it is the simple fact that none of the names present were particularly unique for the time. There are even multiple examples of "Yeshua‘ bar Yehosep” and similar derivations in epigraphy that most certainly doesn't refer to the Jesus. A similar ossuary also exists bearing "Ya'akov bar Yosef achui de Yeshua" ("James, son of Joseph, brother of Jesus") on it, and although there is strong suspicion surrounding it as being a more modern fraud, in any case, even if 'legitimate' it similarly can be explained by the commonality of such names.

In sum, the name was likely a fairly common one in Jewish communities of antiquity, but through the quirks of transliteration, the specific form of 'Jesus' has passed down as being fairly unique and stands out from the Joshuas, even though they wouldn't have at the time, and doesn't in the actual epigraphic evidence of the period.

Sources

Ayalon, Avner, Miryam Bar-Matthews, and Yuval Goren. "Authenticity examination of the inscription on the ossuary attributed to James, brother of Jesus." Journal of Archaeological Science 31, no. 8 (2004): 1185-1189.

Ilan, Ṭal & Kerstin Hünefeld. Lexicon of Jewish Names in Late Antiquity: The Eastern Diaspora 330 BCE-650 CE. Mohr Siebeck, 2002.

Meyers, Eric M. 2006. "The Jesus tomb controversy: An overview". Near Eastern Archaeology 69, (3) (Sep): 116-118,

Oxford English Dictionary. "Jesus" OED Online.

Rollston, Christopher A. "The Talpiyot (Jerusalem) Tombs: Some Sober Methodological Reflections on the Epigraphic Materials" The Jesus Discovery: The New Archaeological Find that Reveals the Birth of Christianity. Simon & Schuster, 2012.

I can give a partial answer; namely, how popular was the name Jesus in Palestine during the Second Temple period (during which Jesus lived)?

Tal Ilan's Lexicon of Jewish Names in Late Antiquity: Part I: Palestine 330 BCE-200 CE (Tubingen: Mohr, 2002) lists the following as the six most popular male names in Palestine during and immediately after the Second Temple period, based on archaeological and literary sources (for example, burial inscriptions, legal documents, the works of historians like Josephus, and Tannaitic [early rabbinic] traditions):

Simon (Shimon): 243 occurrences

Joseph (Yosef): 218 occurrences

Eleazar: 166 occurrences

Judah (Yehudah): 164 occurrences

John (Yochanan): 122 occurrences

Joshua (Yehoshua): 99 occurrences

The name Jesus is an English transliteration of Iesous, a Greek transliteration of the Hebrew name Yeshua (the "us" ending was added to conform to Greek naming conventions for men), which is a diminutive form of the Hebrew name Yehoshua, more commonly transliterated directly into English as Joshua.

R. Hachlili and Richard Bauckham have compiled similar lists, both of which place Yeshua/Yehoshua as the sixth most popular name for a Jewish man in Palestine around the time of Jesus.

Bauckham in Jesus and the Eyewitnesses, 71-74, notes that in general, the names of characters in the Gospels and Acts correlate well with what we would expect given the prevalence of these names in the region at the time.

Certainly these names were drawn from Old Testament characters, but only some of these characters seemed to have been popular choices for names in the Second Temple period, which leads us to ask why these names in particular were chosen so often. Bauckham postulates they seem to have been largely chosen for patriotic reasons. Joshua/Jesus, in particular, may have reflected Messianic expectations, or at the very least, nationalistic ones; the character Joshua in the Old Testament led the conquest of the land of Israel, and there was a strong undercurrent of resistance among first-century Jews coupled with a desire to retake the land from the unwelcome Romans and to reassert political independence. This movement is referenced as the "Zealots" in the New Testament, and it eventually sparked the Jewish Wars beginning in 70 C.E.

Bauckham also notes the conspicuous unpopularity of some names we might expect to have seen: David, Moses, and Elijah. Yet these names were tied so closely to Messianic expectations--the kingly son of David, the prophet like Moses, and Malachi's prophecy of a returning Elijah--that it may have been seen as too presumptuous to assume that one's progeny would in fact fulfill these prophecies or expectations so directly.