The translation choice "memorial" is most unfortunate in modern American usage, as if God is dead so he needs a memorial. Maybe a plaque or a public garden would do.
The OT זכר is better translated by "invocation". It is an active calling out the name, as in Psalm 20:7 (KJV, MT verse 8):
אֵלֶּה בָרֶכֶב וְאֵלֶּה בַסּוּסִים וַאֲנַחְנוּ בְּשֵׁם יְהוָה אֱלֹהֵינוּ נַזְכִּיר
Some trust in chariots, and some in horses: but we will remember the name of the Lord our God
and Exodus 20:24 (KJV, MT 20):
מִזְבַּח אֲדָמָה תַּעֲשֶׂה לִּי וְזָבַחְתָּ עָלָיו אֶת עֹלֹתֶיךָ וְאֶת שְׁלָמֶיךָ אֶת צֹאנְךָ וְאֶת בְּקָרֶךָ בְּכָל הַמָּקוֹם אֲשֶׁר אַזְכִּיר אֶת שְׁמִי אָבוֹא אֵלֶיךָ וּבֵרַכְתִּיךָ
An altar of earth thou shalt make unto me, and shalt sacrifice thereon thy burnt offerings, and thy peace offerings, thy sheep, and thine oxen: in all places where I record my name I will come unto thee, and I will bless thee
In both of these verses, "invoke" would be a better translation in today's American vernacular.
When translating the OT, the KJV consistently follows the Jewish tradition as represented by RASHI. In Psalms 135:13, the KJV and the traditional Jewish interpretation see the two usages of the tetragrammaton as address forms, that is, directly addressing God by his name and praising Him. The logic behind this translation is the previous verse,
And gave their land for an heritage, an heritage unto Israel his people
That is, verse 13 in interpreted to be praise for granting of the land of Sihon and Og and all the kings of Canaan to Israel as an eternal heritage. Because at least part of that land was not included in the promise to Abraham, verse 13 is interpreted as implying eternal title to those lands.
IMHO verse 13 (MT) is more correctly interpreted as an assertion of faith:
יְ֭הוָה שִׁמְךָ֣ לְעוֹלָ֑ם יְ֝הוָ֗ה זִכְרְךָ֥ לְדֹר-וָדֹֽר
"YHVH" is Your name forever, "YHVH" is [the proper form of] Your invocation from generation to generation
This interprets verse 13 as an answer to and affirmation of Exodus 3:15.
The OT parallelism in this verse indicates that "Your name" and "Your [form of ]invocation" (remembrance) are closely related in meaning. The difference between the two is that שם, "name", is a noun with no verb form in Hebrew, whereas זכרך, "invocation", is a noun form that can also be an active verb.