The Gemara in Shabbat states that to fulfill the mitzvah of Hadlakat Neirot for Chanukah on a basic level, a person only needs to light נר איש וביתו, one candle per “household,” regardless of the number of people that reside there. Apparently, the Ikar Mitzvah is keyed to the “home.”
A further indication of the connection between the Mitzvah of Ner Chanukah and the household appears in a Tosafot in Masechet Succah ( מו. ד’’ה הרואה). The Gemara discusses the case where a person walking in the street sees a Menorah. The Gemara says that the person is required to make a Berachah of שעשה ניסים. Tosafot gives two explanations for this unique Halacha: (1) the שעשה ניסים is due to the special excitement around the Mitzvah of Neir Chanukah or (2) it allows one who does not have a home to be able to fulfill some form of the Mitzvah. From the second answer of Tosafot, we can see the ideal way to perform this Mitzvah is when one lights in his home.
As a general rule, most Mitzvot depend on the person performing the Mitzvah, but it seems here that the Mitzvah of lighting candles is in part an “obligation of the home,” not the person, or, as some would say, “it's a דין in the home.”
The Pnei Yehoshua asks, while it makes sense that a Mitzvah such as Mezuzah needs a doorpost or מעקה needs a roof, what does a house have to do with the obligation to light a Menorah? Why is it specifically an obligation connected to a home and not a person?
The Pnei Yehoshua answers that, in fact, the Mitzvah of Neir Chanukah is really a חיוב על הבית. Just like Mezuzah and מעקה needs a house, so does Neir Chanukah.
Rav Yosef Tzvi Rimon (p. 156 in his Sefer on Chanukah) explains that maybe Neir Chanukah is really a Chiyuv on the individual but can only be done in a house. In other words, one of the conditions to fulfill the Mitzvah is to do it in a house. Just like the candles need to burn for a half-hour after צאת הכוכבים, so too the lighting needs to be done in the context of a home.
Rav Yehuda Turetsky (TABC ‘01) quoted a more Machshavah-oriented answer to this question from the Sefat Emet (חנוכה תרל’ד). The Sefat Emet explains that the holiday of Chanukah falls when it is cold and becomes dark outside early. Therefore the role of the candle is to “light up” the physical and emotional darkness. To bring אור to somewhere where it is dark. This is why the Mitzvah of lighting is supposed to be done outside when possible. The Mitzvah of Neir Chanukah represents trying to bring light to somewhere wholly dark, to bring a positive attitude to a cynical, gloomy environment. But this can be quite challenging; how could one do this?
The Sefat Emet explains that it is only difficult if you are alone, but if you act together with a group of people, each person can encourage and support the other. Overcoming challenges are often easier when done with others. Therefore, the house, which symbolizes a family, is the appropriate starting place to light up the darkness. Once a foundation is established with a single candle, we can start adding more per person and all the other Hiddurim. That one light is the most important, and everything else is built based on that.