But the name of the city was "Luz" originally

And he [Ya'aqov] called the name of that place Beis-el, but the name of the city was Luz originally.

- Bereishis 28:19

Luz, the original name for Beis-el, is apparently the name of a kind of tree, usually translated "chestnut". It's one of the kinds of wood from which Ya'aqov avinu made sticks for the sheep and goats to look at while drinking.

Bereishis Rabba (69:8) discusses the amazing properties of living in the city of Luz:

• They always told the truth.
• No one in the city died. When people got old and tired, they needed to move out for nature to take its course.
• The city was never conquered by Sancheirev, and wasn't destroyed by Nevuchadnetzar at the end of the first commonwealth. Even though both invaded Luz.
• Luz is where they made the tekheiles dye.

Luz is also the name of a special bone in the body, where the skull and spine meet. Two medrashim associate the luz bone with Hadrian y"sh. Bereishis Rabba has him trying to grind a luz and failing. There's a strong parallel to the city of Luz resisting conquest at the end of the first beis hamiqdash, since the Hadrianic persecutions are at the end of the second commonwealth. Second, Qoheles Rabba has Hadrian asking R' Yehoshua' ben Chananyah about techiyas hameisim, and RYbC explains that Hashem starts by softening the luz with dew.

(This connection to dew is why the praise of "morid hatal -- He Who lowers dew" is in the berakhah of Shemoneh Esrei that ends "Who revives the dead". It also explains why there is a version in which one says in the summer "morid hatul", with a qamatz, making it the end of the sentence with "mechayei hameisim", while in the summer they would say "umorid hageshem" is with two segol's, connected to "mekhalkeil chaim" -- rain being necessary for this life -- "bechesed"...)

Luz seems particularly connected with Yaiaqov, the one who renames it. First, his service of G-d centers around emes, truth, the middah exemplified by the citizens of Luz. He uses the luz sticks. And according to the Ben Ish Chai, there is a connection to his father-in-law's and brother-in-law's names, as well as his own names/titles.

And the mequbbalim write: There is a bone in a person's body which receives no benefit from food, except from the se'udah revi'is on Motza'ei Shabbos. And this bone does not disintegrate in the grave. It is called variously "niscoi", "luz", and "besu'el". These three names have the acronym of "lavan", which are also the final letters of Yisrael, Yaakov and Yeshurun, and from this bone the body will be rebuilt at techiyas hameisim, and this is specifically applied to Israel only, as the pasuk says: "Ve'atem hadeveqim Bashem E-lokeikhem, chayim kulekhem hayom -- and you who cleave to Hashem your G-d, you are all alive today".

- Ben Ish Chai, yr. 2, Bereishis 27

So, given that Luz was renamed Beis-el, why does the gemara and medrash sometime refer to the city as "Luz"? (Particularly when referring to the city in the times of Sancheirev and Nevuchadnetzar, after many years of it being the Kingdom of Israel's Beis-el.) And what exactly is the common theme here between the tree, the city, the bone and all the people?

The mishnah says "derekh eretz qodmah laTorah -- proper behavior in society is a prerequisite to Torah." Our aggaditos and midrashim seem to converge on underscoring that point. Luz is the city of truth, it has the permanence of truth both territorially and in the lifespans of its inhabitants. And it's truth, the personality trait about which Yaiaqov centers his service of Hashem, which determines techiyas hameisim. All of these medrashim refer to Luz, to the trait. When referring to applying the pursuit of truth to Torah study or worshipping Hashem, then we progress from Luz to Beis-el.

The stick shows the influence of environment. As does the longevity only imparted when one is actually in the city. Luz, the trait, is not a personal endeavor. (Which raises questions of emes vs. shalom, coordinating truth and peace.)

The bone luz is situated just where the mind connects to the body. It is therefore, in a very real sense, "beis keil", G-d's "home" in this world. Ya'aqov builds a circle of stones in which to sleep at this spot, which -- as R' Hirsch notes ad loc -- is the first home of Israel. He gets a vision of a ladder between heaven and earth, an externalized luz bone between mind and body.

Once one has the foundation of "Luz", one has the proper personality and attitude to provide some solidity in time and in social context. Then one is capable of building that derekh eretz into Torah, making their soul a house of G-d.