The estimation of the year of Jesus' birth depends on the estimation of the year of Herod's death. The information to date the latter event is provided by Flavius Josephus in his Antiquities of the Jews, book 17 .
In 17.6.4, when narrating events leading to Herod's last times, he notes an event involving the high priest Matthias ben Theophilus:
Now it happened, that during the time of the high priesthood of this
Matthias, there was another person made high priest for a single day,
that very day which the Jews observed as a fast. The occasion was
this: This Matthias the high priest, on the night before that day when
the fast was to be celebrated, seemed, in a dream, (7) to have
conversation with his wife; and because he could not officiate himself
on that account, Joseph, the son of Ellemus, his kinsman, assisted him
in that sacred office. But Herod deprived this Matthias of the high
priesthood, and burnt the other Matthias, who had raised the sedition,
with his companions, alive. And that very night there was an eclipse
of the moon. (8)
Two important notes regarding this passage.
First, that "very day which the Jews observed as a fast" was specifically Yom Kippur according to rabbinic tradition. Quoting Jewish Encyclopedia:
On the eve of a Day of Atonement—for the priest the most important
time in the year—he had become ritually unclean, and consequently was
unable to perform the duties of his office, which were discharged
instead by his kinsman Joseph ben Ellem ("Ant." xvii. 6, § 4). This
occurrence is mentioned in the Talmud (Tosef., Yoma, i. 4; Yoma 12b;
Yer. Yoma 38d), although the name of Matthias ben Theophilus is
Tractate Yoma in the Tosefta and the Talmuds is all about Yom Kippur services. Quoting Yoma 12b of the Babylonian Talmud:
It happened to Joseph b. Elam of Sepphoris that after a disqualifying
accident had happened to the high priest, he was appointed in the
former's place .
Second, the "very night" when "there was an eclipse of the moon" does not refer to "the night before that day when the fast was to be celebrated", or even necessarily to the night immediately following that, but to the night when "Herod deprived this Matthias of the high priesthood, and burnt the other Matthias, who had raised the sedition, with his companions, alive", which could have happened a few days after the day observed as a fast.
In 17.6.5, when describing Herod's final illness, Josephus notes that Herod "went beyond the river Jordan, and bathed himself in the warm baths that were at Callirrhoe," until "having no longer the least hopes of recovering".
In 17.9.3, when narrating events shortly after Herod's death, Josephus notes "the approach of that feast of unleavened bread, which the law of their fathers had appointed for the Jews at this time, which feast is called the Passover".
Even while the only fast day associated with important priestly duties was Yom Kippur, and moreover, that according to rabbinic tradition the event of a high priest becoming unable to perform his duties and being substituted by a "Joseph, the son of Ellemus" was specifically in Yom Kippur, I will list all Jewish fast days at that time, which were "the fasts of the fourth, fifth, seventh and tenth months" in Zechariah 8:19 plus the Fast of Esther on Purim eve, 13 Adar, instituted after the time of prophet Zechariah (Esther 9:31).
Month No, month name: Fast day, reason.
4, Tammuz: 9 Tammuz, breach of the walls of Jerusalem (2 Kings 25:3-4; Jer 39:2, 52:6–7).
5, Av: 7/10 Av, destruction of the First Temple (2 Kings 25:8-10; Jer 52:12-14).
7, Tishrei: 10 Tishrei, Atonement Day (Yom Kippur) (Lev 16:29-31 and 23:26-28).
10, Tevet: 10 Tevet, beginning of the siege of Jerusalem (2 Kings 25:1-2).
12/13, Adar/Adar II: 13 Adar/Adar II, the Fast of Esther on Purim eve.
Considering all candidate lunar eclipses, we have:
AJ 17.6.4 - Eclipse -- Moon phase - Eclipse date - AJ 17.9.3 --- Jesus' - Jesus'
lunar ---- type and - on day of -- in Hebrew ---- Passover ---- birth -- death
eclipse --- midtime -- eclipse ---- calendar ----- (full moon) - years -- year
 -------  ------  --------  ---------- 
Mar 23, 5 BC - T - 20:21 - Full - 14 Adar II (a) - Apr 11, 4 BC - 7-6 BC - 30 AD
Sep 15, 5 BC - T - 22:12 - Full - 14 Tishrei (b) - Apr 11, 4 BC - 7-6 BC - 30 AD
Mar 13, 4 BC - P - 02:41 - Full - 14 Adar/II (c) - Mar 31, 3 BC - 6-5 BC - 30 AD
Jan 10, 1 BC - T - 01:09 - F.+1 - 15 Shevat (d) -- Apr 07, 1 BC - 4-3 BC - 33 AD
Dec 29, 1 BC - P - 16:32 - Full - 13 Tevet (e) --- Mar 27, 1 AD - 3-2 BC - 33 AD
(a) For Mar 23, 5 BC, the calendar converter  calculates 15 Nisan. However, it is possible that the Hebrew year ending that March was embolismic or leap. Quoting the "Hebrew Calendar" entry in Wikipedia:
During leap years Adar I (or Adar Aleph — "first Adar") is added
before the regular Adar. Adar I is actually considered to be the extra
month, and has 30 days. Adar II (or Adar Bet — "second Adar") is the
"real" Adar, and has the usual 29 days. For this reason, holidays such
as Purim are observed in Adar II, not Adar I. 
Regarding the intercalation of leap years in Herod's time, quoting the "Leap year" entry in Encyclopaedia Judaica:
The intercalation of years was already practiced by the Sanhedrin in
the Hasmonean and mishnaic periods. Among the factors then taken into
consideration were the ripened state of the Omer ("barley") offered on
Passover, and that of the bikkurim ("first fruits") sacrificed on
Shavuot. It also depended on whether the roads and bridges were
passable so that the pilgrims could go to Jerusalem for the Passover
festival, and whether the ovens for the paschal-lamb sacrifices were
already dry after the rainy season. (See: Tosef., Sanh. 2:12; Sanh.
If the Hebrew year corresponding to Julian year 6-5 bC was leap, Mar 23, 5 bC would have been 14 Adar II instead of 15 Nisan. This fits exactly with the lunar eclipse being on the night following the Fast of Esther on Purim eve, 13 Adar II.
The Passover after Herod's death cannot have been the one immediately after this eclipse, which happened on a full moon, since an interval of 29 days is too short for all the events narrated by Josephus between the eclipse and the Passover after Herod's death.
(b) This is consistent with the "very night" when "there was an eclipse of the moon" not referring to the night immediately following the fast of 10 Tishrei, but to the night when "Herod deprived this Matthias of the high priesthood, and burnt the other Matthias, who had raised the sedition, with his companions, alive", with this event happening a few days after 10 Tishrei.
(c) For Mar 13, 4 BC, the calendar converter  calculates 14 Adar II, which would have been 14 Adar if the previous Hebrew year was embolismic or leap as said above. Either case, it fits exactly with the lunar eclipse being on the night following the Fast of Esther on Purim eve, 13 Adar or Adar II.
Just as in the Mar 23, 5 BC eclipse, the Passover after Herod's death cannot have been the one immediately after this eclipse, which happened on a full moon, since an interval of 29 days is too short for all the events narrated by Josephus between the eclipse and the Passover after Herod's death.
Importantly, this eclipse was partial and occurred way too late in the night to be likely to be noted and remembered.
(d) No Jewish fast day was near the date of this eclipse.
(e) This eclipse can be directly discarded because, first, it did not occur at night, and second, but more important, it occurred below the horizon and could not be seen from Jerusalem !
Conclusion: the most probable scenario is the eclipse before the final illness of Herod being on Sep 15, 5 BC, and the Passover after Herod's death being on Apr 11, 4 BC.
Since Herod met the Magi in Jerusalem (Mt 2:1,3), the meeting had to be before he left the city some time after Sep 15, 5 BC. Assuming that the meeting occurred within the 1-year interval Sep 6 BC - Sep 5 BC, and that Herod's subsequent order to execute all babies in the area "two years old and under" reflected his own uncertainty about Jesus' date of birth "in accordance with the time (of the star's appearance) he had ascertained from the magi" (Mt 2:7,16), Jesus could have been born between 2 years before the start and 6 months before the end of that interval, i.e. between Sep 8 BC - March 5 BC.
 Antiquities of the Jews - Book XVII
 Jewish Encyclopedia
 Babylonian Talmud, Tractate Yoma
 NASA. "Catalog of Lunar Eclipses: -0099 to 0000 (100 BCE to 1 BCE)". Online at: NASA Eclipse Web Site
(On that page, 1 bC = 0000, 2 bC = -0001, and 4 bC = -0003.)
 Phases of the Moon: -0099 to 0000 (0100 to 0001 BCE)
 Calendar Converter
 United States Naval Observatory (USNO). "Spring Phenomena, 25 BCE to 38 CE". Online at: Spring Phenomena 25 BCE to 38 CE (lately available only through google cache, accessed searching for the page title.)
 Hebrew calendar
 LEAP YEAR - Jewish Virtual Library