Although I am not aware of this quote, "something similar" might be Deuteronomy (25:6) which states that in the event that a man dies childless, his brother is to marry his widow, "and the firstborn son she gives birth to will stand in his brother's name, and his name shall not be wiped out from Israel" (my rough translation). Some pashtanim interpret this as instructing the living brother to name the son he has with his deceased brother's widow after his deceased brother (Ibn Caspi and Ralbag there, implication of Abravanel to Deut. 25 as noted by HaKtav V'HaKabbalah to 25:6, and note Ibn Ezra). This has to do with perpetuating the deceased brothers' legacy (Ralbag to 25:5-6). According to this explanation, perpetuating the name of the deceased is considered like perpetuating him.
Interestingly, the destruction of Amalek includes destroying זכר עמלק (Deut. 25:19); meaning the memory or mention of Amalek (that the same word can be used for both is itself significant). Some interpret this as an instruction to destroy their property so that their name will no longer me mentioned in connection with their former property (Rashi there).
Interestingly, Proverbs (10:7) which speaks of the name of the wicked rotting, is interpreting by the Talmud (Yoma 38b) as instructing people to not name people after the wicked.
These are admittedly loose connections.
Significantly, after this question, the first google result for related terms, is project stolpersteine, and after some searching I cannot find such a quote in the Talmud. Also notable is that such sentiments are found in other recent sources (listed here). For example, David Eagleman writes:
There are three deaths. The first is when the body ceases to function. The second is when the body is consigned to the grave. The third is that moment, sometime in the future, when your name is spoken for the last time.