For Further Iyun
1. Concerning whether construction itself actually began on the 24th or 25th of Kislev, note (see 2:15) which indicates that construction was to begin the next day:

"On the 24th day of the ninth [month], in the second year of Daryavesh (Darius), the word of the Lord came to the prophet Chagai ... 'And now, take note from this day forward, as long as no stone has been laid on another in the House of the Lord ... for from this day on I will send blessings...' " (see 2:10-19)

2. In his article "Yom Yisud Heikhal Hashem," Megadim Vol. 12, Rav Yoel Bin-Nun addresses this issue at length. His approach emphasizes the agricultural importance of this date, marking the end of the olive harvest, and the finale of the agricultural season of the previous year.
He also deals with the historical importance of this date and its relationship to Chagai and Zecharya, as was presented in this shiur in a more simplified manner.

3. Chayil can also imply economic prosperity; see Devarim 8:17 in its context. This understanding would relate nicely to the prophecy of Chagai.

4. See Macabees I chapter 4:36-60. From this account, it appears that more than one day was necessary to prepare the mikdash for the dedication ceremony on the 25th of Kislev. It was necessary to build a new mizbayach, shulchan, menorah, etc., and to prepare the courtyards, as well as remove all the pagan idols.

(See 1:54-61 of Sefer Ha-makkabim I, part of the Apocrypha. Although this book is not prophecy, from its style it is clear than the author was a god-fearing Jew. See introduction by Avraham Kahane "Ha-Sefarim Ha-Chitzonim" [Hebrew].)

5. See Mesechet Avoda Zara 8a, the first gemara after the Mishneh. It suggests that the time of year of Chanuka had already carried significance since the time of Adam ha'Rishon!

6. The "semel ha'Medina" - Israel's national symbol - is modeled after Zecharya's menorah (see 4:1-9 and the Knesset Gounds). In your opinion, based on the nevuot of Zecharya, was this a good choice?