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To my knowledge, there are 3 mitzvot in the Torah that explicitly state that the reward is lengthened days (long life):

1) Honoring one's parents (Exodus 20:12)

כַּבֵּ֥ד אֶת־אָבִ֖יךָ וְאֶת־אִמֶּ֑ךָ לְמַ֙עַן֙ יַאֲרִכ֣וּן יָמֶ֔יךָ עַ֚ל הָאֲדָמָ֔ה אֲשֶׁר־יְהוָ֥ה אֱלֹהֶ֖יךָ נֹתֵ֥ן לָֽךְ׃

Honor your father and your mother, so that your days be lengthened on the land that the LORD your God is assigning to you.

2) Sending away the mother bird (Deuteronomy 22:7)

שַׁלֵּ֤חַ תְּשַׁלַּח֙ אֶת־הָאֵ֔ם וְאֶת־הַבָּנִ֖ים תִּֽקַּֽח־לָ֑ךְ לְמַ֙עַן֙ יִ֣יטַב לָ֔ךְ וְהַאֲרַכְתָּ֖ יָמִֽים׃

Let the mother go, and take only the young, in order that you may fare well and your days be lengthened.

3) Honest weights (Deuteronomy 25:15)

אֶ֣בֶן שְׁלֵמָ֤ה וָצֶ֙דֶק֙ יִֽהְיֶה־לָּ֔ךְ אֵיפָ֧ה שְׁלֵמָ֛ה וָצֶ֖דֶק יִֽהְיֶה־לָּ֑ךְ לְמַ֙עַן֙ יַאֲרִ֣יכוּ יָמֶ֔יךָ עַ֚ל הָֽאֲדָמָ֔ה אֲשֶׁר־יְהוָ֥ה אֱלֹהֶ֖יךָ נֹתֵ֥ן לָֽךְ׃

You must have completely honest weights and completely honest measures, so that your days be lengthened on the soil that the LORD your God is giving you.


Is there a common denominator as to why specifically these 3 mitzvot reward lengthened days?

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    Devarim 11:21 - לְמַ֨עַן יִרְבּ֤וּ יְמֵיכֶם֙ וִימֵ֣י בְנֵיכֶ֔ם עַ֚ל הָֽאֲדָמָ֔ה אֲשֶׁ֨ר נִשְׁבַּ֧ע יְהוָ֛ה לַאֲבֹתֵיכֶ֖ם לָתֵ֣ת לָהֶ֑ם כִּימֵ֥י הַשָּׁמַ֖יִם עַל־הָאָֽרֶץ׃ (ס) to the end that you and your children may endure, in the land that the LORD swore to your fathers to assign to them, as long as there is a heaven over the earth. sefaria.org/Deuteronomy.11.21?lang=bi&with=all&lang2=en – Salmononius2 May 13 '19 at 15:28
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    don't forget, "Kiddushin 39b:7-8 re: honouring parents, it says, "so your days will be lengthened, and so it will be good for you" (Deut. 5:15). Re: sending away the mother bird, it says, "so it will be good for you and your days will be lengthened" (Deut 22:7). For when his father says to him, "Go up and bring me chicks!", he goes up, sends away the mother [bird], takes the children, but on return, falls and dies. Where are lengthening of this one's days? But, "so that it be good for you"—in the world which is all good; "And so that your days will be long"—in the world which is all long. – Avrohom Yitzchok May 13 '19 at 16:13
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    Devarim 4:40 5:30 6:2 11:9 17:20 30:20 32:47 – Double AA May 13 '19 at 19:07
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    Learning Torah - כי הוא חייך ואורך ימיך? – DonielF May 13 '19 at 20:16
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    A king should keep a Sefer Torah by his side למען יאריך ימים על ממלכתו – DonielF May 13 '19 at 20:22
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To me it always seemed the common denominator is that:
things that take time, will give you more time,
Or in other words:
When you give G-d Almighty your time he will surely repay you.

(all these mitzvos demand (a lot of) patience,
(with this information (that you will have more time) you can tell your yetzer hara do not worry you are not loosing your time))

Honoring parents takes time, (you care for them instead of caring for yourself),.
Sending away the bird (when you easily can take it): makes you need more time to catch your meal.
Not cheating (when you easy can): makes you need more time to make money (for your sustenance)


Learning Torah: takes time

Maybe it is not only for the mezuzah that you get long life, it is for all the Mitzvos listed right before (most of which take time/patience)):
Therefore impress these My words upon your very heart: bind them as a sign on your hand and let them serve as a symbol on your forehead, and teach them to your children—reciting them when you stay at home and when you are away, when you lie down and when you get up; and inscribe them on the doorposts of your house and on your gates

Or maybe it just has to do with G-d's name in the mezuzah (Menachot 44b) (not just using your time to honor G-d Almighty, (to hang a Mezuzah takes time, and patience), but also honoring connecting to G-d (who is good) who is life, directly, must give you life.
(This gemmora might also be the Lubacher Rebbe's source for checking the Mezuzah and Teffillin for health)

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