From what I understand the Well of Miriam in the desert was actually a rock which miraculously brought forth water. As such why do we refer to it as the 'Well of Miriam' and not the 'Rock of Miriam'?
My guess is that there was nothing special about the rock, which existed way before Miriam and lasted long after her. The only thing special about it was that it produced water like a well, which lasted only as long as Miriam was alive.
Hence, it wasn't the rock of Miriam, since she had nothing to do with it being a rock. It is the well of Miriam, since she caused it to be a well.
In Bamidbar 21 it's referred to as a well, multiple times:
טז וּמִשָּׁם, בְּאֵרָה: הִוא הַבְּאֵר, אֲשֶׁר אָמַר יְהוָה לְמֹשֶׁה, אֱסֹף אֶת-הָעָם, וְאֶתְּנָה לָהֶם מָיִם. יז אָז יָשִׁיר יִשְׂרָאֵל, אֶת-הַשִּׁירָה הַזֹּאת: עֲלִי בְאֵר, עֱנוּ-לָהּ. יח בְּאֵר חֲפָרוּהָ שָׂרִים, כָּרוּהָ נְדִיבֵי הָעָם, בִּמְחֹקֵק, בְּמִשְׁעֲנֹתָם; וּמִמִּדְבָּר, מַתָּנָה
My guess is that as a result it's called the Well of Miriam, though the Ibn Ezra on that Pasuk says that it does not refer to Be'er Miriam.
וזאת הבאר גם היא היתה פלא ואיננה הבאר הנקרא באר מרים לפי דעתי
Perhaps because it wasn't just a rock - first and foremost it served the function of a well.
...בארה של מרים - סלע וזבין ממנו מים והיה מתגלגל והולך עם ישראל
The Well of Miriam - a rock and water flowed from it and it would roll and (lit.) go (i.e. travel) with the Jewish People.
So it would seem whilst it was indeed a rock, it was a miraculous feature within the Jewish peoples' existence in the desert. It provided a wellspring of water and would travel in tandem with them.
In fact, we can build on this by bringing another Gemara in Shabbos 35a which shows that despite being a rock its modus operandi was to provide a constant stream of water and therefore function as a well.
It writes there:
אָמַר רַבִּי חִיָּיא: הָרוֹצֶה לִרְאוֹת בְּאֵרָהּ שֶׁל מִרְיָם יַעֲלֶה לְרֹאשׁ הַכַּרְמֶל וְיִצְפֶּה וְיִרְאֶה כְּמִין כְּבָרָה בַּיָּם — וְזוֹ הִיא בְּאֵרָהּ שֶׁל מִרְיָם. אָמַר רַב: מַעְיָן הַמִּיטַּלְטֵל טָהוֹר — וְזֶהוּ בְּאֵרָהּ שֶׁל מִרְיָם
Rabbi Ḥiyya said: One who wants to see Miriam’s well, which accompanied the Jewish people throughout their sojourn in the desert, should do the following: He should climb to the top of Mount Carmel and look out, and he will see a rock that looks like a sieve in the sea, and that is Miriam’s well. Rav said: A spring that is portable, i.e., that moves from place to place, is ritually pure and is regarded as an actual spring and not as drawn water. And what is a movable spring? It is Miriam’s well. (Sefaria translation & notation)