PART 1
I've been doing some reasonably intensive research on what ancient sources thought about Jesus' dates, as it happens, so I have a lot of the information at my fingertips.
It's ancient Christian writers and chronographers that arrived at 1 BCE (well, some of them, anyway) as Jesus' birth date. As you mention, the modern 'consensus' gravitates towards an earlier date.
The 1 BCE date begins to appear in Christian chronographical thought in the mid-200s CE, in the pseudo-Cyprianic De Pascha computus 18-23 which dates to 243, and later in Eusebius' Armenian chronicle (early 300s), and the Chronography of 354, actually based on material dating to 336, in the 'fasti consulares' section.
It's popularly repeated that the earliest use of an 'anno Domini' dating system only began to be used in the 500s, with Dionysius Exiguus, but really that's either too late or too early: too late, because the 1 BCE date is already there in the sources I've mentioned, or too early, because 'A.D.' didn't take off as a calendar-era system until people like Alcuin in the 8th century.
Now, if you think the 200s seems pretty late for such a precise dating ... you'd be right.
##What early Christians thought about Jesus' dates
This bit isn't simple. It's perfectly transparent from Christian sources of the first two centuries that no extant writers had access to any information beyond what's in the gospels, specifically the opening chapters of Matthew and Luke. They're the only primary sources that ancient Christians had. Everything we see in later writers, even when they start quoting hyper-specific dates for Jesus' birth and death, rests on nothing firmer than those two texts.
And from a modern perspective, that's a problem, because Matthew and Luke quote four sets of chronological markers, and most of them disagree with each other.
These chronological markers span at least a decade. That could be worse, by the standards of ancient history, but it's certainly not precise.
We have four other 1st-2nd century sources that comment on Jesus' dates: Josephus, Tacitus, Justin Martyr, and Irenaeus. They contain no information beyond beyond what's in the gospels. Literally all they say is that Jesus died during Tiberius' reign, as in Luke 3. Irenaeus repeats this in three different places, in contexts where he's clearly trying to make a point about chronological precision, and that makes it clear that it's the limit of his information.
[Edit, four months later: actually it's Jesus' baptism that Irenaeus puts in Tiberius' reign without further specification; he thinks Jesus' ministry lasted more than a decade, and puts his death in Claudius' reign.]
And yet just a few years later, around 200 CE, we suddenly find Clement of Alexandria (Stromateis 1.21.144-146) discussing datings for Jesus that are precise to the day.
Hmmm.
Is it possible that independent traditions about Jesus' dates existed? Could Clement and later writers have had access to more precise information via oral tradition?
Well ... no, not really. Like I said, Irenaeus is pretty clear that he's giving info about Jesus' dates to the maximum precision that he had. If there was some independent tradition floating around, then it's a tradition that basically no one had access to. Not exactly a well disseminated tradition, then! No, the simplest reading is that Josephus, Tacitus, Justin, and Irenaeus had no information other than what we see in the gospels. (Josephus may have had access to an earlier form of this information: it's moderately likely that Luke is later than Josephus. If so, it's still no more precise.)
##How then, did ancient Christians come up with a date of 1 BCE?
The synoptic gospels (Mark, Matthew, and Luke) refer to a darkness at the time of Jesus' death which lasted from midday until the ninth hour, that is, about three hours. Starting in the 100s, some people started trying to link this darkness to a solar eclipse. The earliest is Thallos, whose work is lost (BNJ 256 F 1). Tertullian (ca. 200), Julius Africanus (220s), Origen (ca. 250), and Eusebius (early 300s) all discuss this interpretation, and so do several later writers.
By the 220s people were trying to tie this darkness to a specific eclipse, one that took place in 29 CE. The information they had about the 29 CE eclipse came from a pagan writer, Phlegon of Tralles, who apparently reported it in a lost chronographical work called the Olympiads (BNJ 257 T 16a-e).
Their information wasn't reliable, of course. There are many problems. The biggest two are:
1. Calendar-era systems. All the sources are using different calendar-era systems, that is to say systems for expressing which year an event took place. The Roman empire didn't have a single calendar-era system. In Rome you could use consular fasti, and name the consuls in that year; under the principate, an alternate system took hold, referring to how many tribunates the emperor had held. (And just to be convenient, the tribunician year began on 10 December, not 1 January.) But in other parts of the empire many other systems were used. The most common system in the east was to refer to the emperor's regnal year -- but regnal years began at different times of year depending on where you were. Other systems that we find in sources on Jesus' dates include Olympiads (bunches of four years, starting from the 1st Olympiad in 776-771 BCE, with each year running from midsummer to midsummer), years since Abraham, years since Adam, and others. (And by the way, when Julius Africanus refers to a number of years since the Creation, his count is different from that used by contemporary Jews and the official Byzantine year count.)
There are nice convenient calendar-era systems that they could have used. But they don't. Systems like 'AUC' (counting years from the legendary founding of Rome: unfortunately that's mostly a modern fad, no one used it in antiquity), or the Seleucid year (which was in widespread use in the Levant until the late Mediaeval period). Ancient Christian chronographers didn't want things to be simple.
2. The 29 CE eclipse couldn't possibly have coincided with Jesus' death. Julius Africanus sensibly points out (Chronographiae F 93 ed. Wallraff) that Jesus died at Passover, but Passover is at full moon, while solar eclipses happen at new moon. In addition, we can add that totality lasted less than two minutes, not three hours; the path of totality passed about 700 km north of Jerusalem; and the eclipse took place in November, not in spring.
It's pretty clear that the Christian sources on Jesus' dates didn't have direct access to Phlegon's report of the eclipse. Origen didn't know which bit of Phlegon's book talked about the eclipse, and he gets the title wrong. Africanus says Phlegon talked about a three-hour eclipse, but that's obviously contamination from the gospels, and that means that Africanus' source had that contamination. (They didn't have direct access to consular fasti either: when they name the consuls at the time of Jesus' death, they get the correct consuls for 29 CE, but without exception, every ancient Christian source misspells their names, in a variety of ways.)
Be that as it may, an in spite of some sources' scepticism about the eclipse, the 29 CE eclipse seems to have become the linch pin of ancient Christian efforts to pinpoint Jesus' dates. The argument runs:
There are additional problems I'm glossing over here. Eusebius' chronology puts the eclipse in 32/33 CE, for example -- but I think it'd be too much of a distraction to go into that here.
[Part 2 follows below]
I commend to your attention the appropriate section of the FAQ. Should your browser not take you there automatically, that's 'How did the world agree on what year it is?', specifically the first and last answers linked there.