When considering any topic, I find it valuable to consider Who, What, When, Where, Why, and How. In the Babylonian Talmud, the early rabbis (tannaim) discuss many aspects of Hanukah lights, which we will explore within a framework of the 5 Ws and How.
Who? & What? (& How?)
תנו רבנן:
מצות חנוכה נר איש וביתו
והמהדרין נר לכל אחד ואחד
והמהדרין מן המהדרין:
בית שמאי אומרים יום ראשון מדליק שמנה מכאן ואילך פוחת והולך
ובית הלל אומרים יום ראשון מדליק אחת מכאן ואילך מוסיף והולך
Our Rabbis taught:
The mizvah of Hanukah is one candle per household.
For the beautifiers: one candle per person.
For the beautifiers of the beautifiers:
Shammai’s Academy says: “On the first day, light eight; thenceforth, decrease.”
Hillel’s Academy says: “On the first day, light one; thenceforth, increase.”
When?
מצותה משתשקע החמה עד שתכלה רגל מן השוק
Its mizvah is from sunset until traffic in the marketplace ceases.
Where?
תנו רבנן:
נר חנוכה מצוה להניחה על פתח ביתו מבחוץ
אם היה דר בעלייה מניחה בחלון הסמוכה לרשות הרבים
ובשעת הסכנה מניחה על שלחנו ודיו
Our Rabbis taught:
It is a mizvah to place the Hanukkah lamp by the door of one’s house on the outside;
if one dwells in an upper chamber, one places it at the window nearest the street.
But in times of danger, it is sufficient to place it on the table.
Thus far, there is an absence of Why? There is one text that comes very close to answering this question:
תנו רבנן בכה בכסליו יומי דחנוכה תמניא אינון דלא למספד בהון ודלא להתענות בהון שכשנכנסו יוונים להיכל טמאו כל השמנים שבהיכל וכשגברה מלכות בית חשמונאי ונצחום בדקו ולא מצאו אלא פך אחד של שמן שהיה מונח בחותמו של כהן גדול ולא היה בו אלא להדליק יום אחד נעשה בו נס והדליקו ממנו שמונה ימים לשנה אחרת קבעום ועשאום ימים טובים בהלל והודאה
Our Rabbis taught:
On the twenty-fifth of Kislev, [commence] the days of Hanukkah, which are eight on which a lamentation for the dead and fasting are forbidden.
For when the Seleucids entered the Temple, they defiled all the oils therein, and when the Hasmonean dynasty prevailed against and defeated them, they made search and found only one cruse of oil which lay with the seal of the High Priest, but which contained sufficient for one day’s lighting only; yet a miracle was wrought therein and they lit [the lamp] therewith for eight days.
The following year, these [days] were appointed a Festival with [the recital of] Hallel and thanksgiving.
This text could have ended: “The following year, they established them and made them festive days with Hallel, thanksgiving, and lighting of candle(s)”, but, since it didn’t, it particularly stands out. This lack of a particular reason proffered as to how and why the sages decided to institute lighting of candles is, of course, quite curious.
Nevertheless, there is a value that a later rabbi attaches to the lighting of Hanukah candle(s), as we see occurring in the early fourth century:
אמר רבא פשיטא לי נר ביתו ונר חנוכה נר ביתו עדיף משום שלום ביתו
נר ביתו וקידוש היום נר ביתו עדיף משום שלום ביתו
בעי רבא נר חנוכה וקידוש היום מהו קידוש היום עדיף דתדיר או דילמא נר חנוכה עדיף משום פרסומי ניסא
בתר דאבעיא הדר פשטה נר חנוכה עדיף משום פרסומי ניסא:
Rava said: “It is obvious to me [that if one must choose between] the house light and the Hanukah light, a house light is preferable, for peace of the home.
[Between] the house light and sanctification of the day – the house light is preferable, on account of the peace of the home.”
Rava asked: “What about [choosing between] a Hanukah candle and sanctification of the day? Is it that sanctification of the day is preferable, since it occurs with more frequency or, perhaps, the Hanukah candle is preferable on account of publicizing of the miracle?”
After he asked, he then went back and laid it out: “A Hanukah candle is preferable on account of the publicizing of the miracle.”
The seeming lack of the rabbis to ascribe a particular reason to the lighting of the Hanukah candle(s) brings to mind a first century author who also did not know the origins of the lighting:
Josephus: Antiquities of the Jews 7: 323-326 [93-93 CE]
Now Yehudah celebrated the festival of the restoration of the sacrifices of the temple for eight days; and omitted no sort of pleasures thereon: but he feasted them upon very rich and splendid sacrifices; and he honored God, and delighted them, by hymns and psalms. Nay, they were so very glad at the revival of their customs, when after a long time of intermission, they unexpectedly had regained the freedom of their worship, that they made it a law for their posterity, that they should keep a festival, on account of the restoration of their temple worship, for eight days. And from that time to this we celebrate this festival, and call it 'Lights'. I suppose the reason was, because this liberty beyond our hopes appeared to us; and that thence was the name given to that festival.
Going back earlier, we can try exploring what was being celebrated
1 Maccabees 4 NRSV
41: Then Judas detailed men to fight against those in the citadel until he had cleansed the sanctuary. 41. He chose blameless priests devoted to the law, 43 and they cleansed the sanctuary and removed the defiled stones to an unclean place. 44 They deliberated what to do about the altar of burnt offering, which had been profaned. 45 And they thought it best to tear it down, so that it would not be a lasting shame to them that the Gentiles had defiled it. So they tore down the alter, 46 and stored the stones in a convenient place on the temple hill until a prophet should come to tell what to do with them. 47 Then they took unhewn stones, as the law directs, and built a new altar lik the former one. 48 They also rebuilt the sanctuary and the interior of the temple, and consecrated the courts. 49 They made new holy vessels, and brought the lampstand, the altar of incese, and the table into the temple. 50 Then they offered incense on the altar and lit the lamps on the lampstand, and these gave light in the temple... 56 So they celebrated the dedication of the altar for eight days, and joyfully offered burnt offerings; they offered a sacrifice of well-being and a thanksgiving offering... 59 Then Judas and his brothers and all the assembly of Israel determined that every year at that season the days of dedication of the altar should be observed with joy and gladness for eight days, beginning with the twenty-fifth day of the month of Kislev.
Here, in I Maccabees, we read of the rededication of the Temple, including the lamps in the menorah, although no obligation/practice for Jews outside of the Temple to do so.
II Maccabees 10:1 - 9 (translation from the Oxford Standard Revised Edition of the Bible)
Now Maccabeus and his followers, the Lord leading them on, recovered the temple and the city; they tore down the altars that had been built in the public square by the foreigners and also destroyed the sacred precincts. They purified the sanctuary, and made another altar of sacrifice; then, striking fire out of flint, they offered sacrifices, after a lapse of two years, and they offered incense and lighted lamps and set out the bread of the Presence. When they had done this, they fell prostrate and implored the Lord that they might never again fall into such misfortunes, but that, if they should ever sin, they might be disciplined by him with forbearance and not be handed over to blasphemous and barbarous nations. It happened that on the same day on which the sanctuary had been profaned by the foreigners, the purification of the sanctuary took place, that is, on the twenty-fifth day of the same month, which was Kislev. They celebrated for eight days with rejoicing, in the manner of the festival of booths, remembering how not long before, during the festival of booths, they had been wandering in the mountains and caves like wild animals. Therefore, carrying ivy-wreathed wands and beautiful branches and also fronds of palm, they offered hymns of thanksgiving to him who had given success to the purifying of his own holy place. They decreed by public edict, ratified by vote, that the whole nation of the Jews should observe these days every year.
In II Maccabees, we read of a similar telling: that of the rededication of the Temple, including the lamps in the menorah, but, again, no mention of lighting candles outside of the Temple.
Here is one final source to consider
תנו רבנן:
לפי שראה אדם הראשון יום שמתמעט והולך אמר אוי לי שמא בשביל שסרחתי עולם חשוך בעדי וחוזר לתוהו ובוהו וזו היא מיתה שנקנסה עלי מן השמים
עמד וישב ח' ימים בתענית [ובתפלה]
כיון שראה תקופת טבת וראה יום שמאריך והולך אמר מנהגו של עולם הוא
הלך ועשה שמונה ימים טובים
לשנה האחרת עשאן לאלו ולאלו ימים טובים הוא קבעם לשם שמים
והם קבעום לשם עבודת כוכבים
The Sages taught:
When Adam the first man saw that the day was progressively diminishing, said: "Woe is me; perhaps because I sinned, the world is becoming dark around me and will ultimately return to the primordial state of chaos and disorder."And this is the death that was sentenced upon me from Heaven."
He arose and spent eight days in fasting and in prayer.
Once he saw that the season of Tevet, i.e., the winter solstice, had arrived, and saw that the day was progressively lengthening after the solstice, he said: "This the order of the world."
He went and observed a festival for eight days.
Upon the next year, he observed both these eight days on which he had fasted on the previous year, and these eight days of his celebration, as days of festivities. He, Adam, established these festivals for the sake of Heavenץ
But they, the gentiles of later generations, established them for the sake of idol worship.