From what I understand the Well of Miriam in the dessert 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