One of the letters is from a Cedarhurst rabbi who defends the sanctity of Yiddish, citing Maimonides:
And finally, to refer to the Yiddish language as not a “holy” language, but just another example of clinging to the world of the exiled and battered Jew, is doing a great disservice to the countless Jews in today’s world who view the Yiddish language as sacred. And by the way, I believe so do Chazal. The Rambam, in his Introduction to the Mishnah questions why the Gemara was written in Aramaic and not in Lashon HaKodesh. After all, the Jews learned that language during galus Bavel; why would they write the Torah down in this language that is a “galus language”? And the Rambam explained that since the great Sages of the time spoke in Aramaic in their daily Torah studies, it became elevated and has a din of a lashon ha’kodesh. Yiddish—spoken for hundreds of years by the leading sages of each generation and passed on from rebbi to talmid and from father to son—clearly can be referred to as a lashon ha’kodesh. True, it does not have a level of kedushah of Lashon haKodesh itself, but it has earned the right to be referred to as a holy language, as well.
However, I find this puzzling, as try as I may I can't find the Rambam's discussion of Aramaic in his Introduction to the Mishnah.
What I did find was his discussion of why the post-Talmudic Sages write works in Arabic and Hebrew which explain the Talmud:
וכאשר מתו כל החכמים ע"ה שהאחרונים מהם רבינא ורב אשי וכבר נשלם התלמוד, הרי כל מי שעמד אחריו אן מטרתו אלא הבנת דבריהם שחברו בלבד, עליו אין להוסיף וממנו אין לגרוע, ולפיכך חברו הגאונים הפירושים המרובים, אבל לפי ידיעתנו לא יכל אף אחד מהם להשלים פירוש כל התלמוד, יש שמנעו קוצר החיים, ויש שמנעתו טרדת בני אדם במשפטיהם. וכן חברו חבורים בפסקי הלכות מהם בערבית ומהם בעברית, כגון הלכות גדולות, והלכות קטועות, והלכות פסוקות, והלכות רב אחא משבחא, וזולתם
As far as I can tell Aramaic is not mentioned even once in the Introduction to the Rambam's Commentary on the Mishnah; certainly he does not say or imply that Aramaic is "elevated and has a din of a lashon ha’kodesh," or anything like it.
The only other conceivable reference is to Hebrew and less-than-stellar knowledge of it is
והיה צח לשון ובקי בשפה העברית יותר מכל אדם, עד שהיו החכמים ע"ה לומדין ביאור מלים שנסתפקו להם בלשון המקרא מפי עבדיו ומשרתיו, וזה מן המפורסמות בתלמוד
which lauds Rabbi's expertise in Hebrew (albeit with a limitation, discussed here).
I think this is an example of how a popular idea might be projected onto a text and a person without double checking the source.
Interestingly, the Rambam himself does not agree that the reason why Hebrew was sometimes called לשון הקודש i is because it was "spoken for hundreds of years by the leading sages of each generation and passed on from rebbi to talmid and from father to son" or that "the great Sages of the time spoke in [Hebrew] in their daily Torah studies."
Rather, Rambam believed that Hebrew can be called לשון הקודש because
I have also a reason and cause for calling our language the holy language-do not think it is exaggeration or error on my part, it is perfectly correct-the Hebrew language has no special name for the organ of generation in females or in males, nor for the act of generation itself, nor for semen, nor for secretion. The Hebrew has no original expressions for these things, and only describes them in figurative language and by way of hints, as if to indicate thereby that these things should not be mentioned, and should therefore have no names; we ought to be silent about them, and when we are compelled to mention them, we must manage to employ for that purpose some suitable expressions, although these are generally used in a different sense. Thus the organ of generation in males is called in Hebrew gid, which is a figurative term, reminding of the words, And thy neck is an iron sinew" (gid) (Isa. xlviii. 4). It is also called shupka, pouring out 'I (Deut. xxiii. 2), on account of its function. The female organ is called kobah (Num. xxv. 8), from kebab (Dent. xviii. 3), which denotes" stomach": rehem," womb," is the inner organ in which the foetus develops; zoah (Isa. xxviii. 8)," refuse," is derived from the verb yaza," he went out"; for" urine" the phrase meme raglayim," the water of the feet" (2 Kings. xviii. 17), is used; semen is expressed by shikbat zera'," a layer of seed." For the act of generation there is no expression whatever in Hebrew: it is described by the following words only: ba'al," he was master": shakab," he lay": lakah," he took"; gillah 'ervah," he uncovered the nakedness." Be not misled by the word yishgalennah (Deut. xxviii. 30), to take it as denoting that act: this is not the case, for shegal denotes a female ready for cohabitation. Comp." Upon thy right hand did stand the maiden" (shegal)" in gold of Ophir" (Ps. xlv. io). Yishgalennah, according to the Kethib, denotes therefore he will take the female for the purpose of cohabitation."
Guide to the Perplexed 3:8 (Friedlander)
Incidentally, I side with the Yiddishists--in the sense that it is nothing shameful about our past and present, but part of our rich cultural and religious heritage.
 There are, in fact, only a few scattered Talmudic and Midrashic references to Hebrew by the לשון הקודש appellation, which became much popular later.