Prior to the Mabul people lived for hundreds of years. After the Mabul lifespans shortened although we see that the Avos lived in the 180 year range. In the Pasuk in Tehilim we say Yemei Shenoseinu Shivim Shana, V'Im Bgevuros Shemonim Shana. Where does the saying "You should live till 120" come from? And why Davka 120?
The Torah writes about Moshe that he remained youthful and vigorous until 120 years. We thus bless each other with the designation "Until 120" with the same connotation, viz. that they should live a long life without any physical, emotional and intellectual degradation.
וּמֹשֶׁה, בֶּן-מֵאָה וְעֶשְׂרִים שָׁנָה--בְּמֹתוֹ; לֹא-כָהֲת עֵינוֹ, וְלֹא-נָס לֵחֹה.
And Moses was a hundred and twenty years old when he died: his eye was not dim, nor his natural force abated.
Although there are a number of people who lived past 120 years, only Moshe is described as living a long life without ever experiencing the ravages of old age.
I also used to think it had to do with Moshe's age. But then this made me wonder why we say "You should live till 120", since it sounds more like a curse than a blessing to limit someone's potential age to a set number of years. Then it was pointed out to me that it isn't a curse, but a determination made by HaShem, explicitly stated in the Torah.
In BeReishith (6:3) HaShem says: "לֹא-יָדוֹן רוּחִי בָאָדָם לְעֹלָם, בְּשַׁגַּם, הוּא בָשָׂר; וְהָיוּ יָמָיו, מֵאָה וְעֶשְׂרִים שָׁנָה. "
"My spirit shall not abide in man for ever, for that he also is flesh; therefore shall his days be a hundred and twenty years."
Essentially, we are saying that the person should live to the maximum age he can, which is set at 120 years as ordained by HaShem.
Rashi holds that the Pasuk is referring to something else, but I believe that is likely where the tradition stems from. Rashi's opinion is not shared by everyone.
An answer is suggested by the Gemara in Chulin 139b which asks: Moshe min ha-Torah me-nayin? (Where is Moshe alluded to in the Torah?) in which Moshe is linked to the phrase in Beresheet that human life not exceed 120 years. Even though the plain meaning there is that the flood was to arrive in 120 years (wiping out humanity save Noah and family) the accompanying explanation (6:3)given is that humans are made of flesh. Again, the plain meaning is that flesh is the cause of their corruption and the coming flood. we can conclude though we our longevity is limited by our physicality--as flesh beings 120 is a maximum. It was only because Moshe was flesh that he would die at 120--although he had attained an enlightened state and his body reflected light, he was still in a body. May you live to 120.
To be complete: Perhaps the tradition emerged since Moshe is the final person mentioned in the Torah and a paradigm of (near) perfection and as stated above was connected to the phrase of limited longevity for the human body.
It's a common mistake.
The meaning of the Pasuk you cited is that Hashem would wait 120 years before the punishing that generation (Rashi). So "till 120" hasn't came from there, if there is any source at all