A 17th century Hebraist imagines how rabbis receive their titles.
An illustration of this point can be made by perusing a very interesting excerpt from a 1678 English book called Moses and Aaron: Civil and Ecclesiastical Rites, Used by the Ancient Hebrews by Thomas Goodwin. (Note his reliance on the 10th century Arukh  of R. Nathan ben Yechiel of Rome)
As you can see, on the second page Goodwin illustrates the naming principle he just outlined using the example of Maimonides and Gersonides: "Maimonides, at first was termed only Ben Maimon, the son of Maimon, after his degree, then was he called by his own name, added to his fathers, Moses Ben Maimon, Moses the son of Maimon: at last being licenses to teach, then was he called רמבם Rambam, which abbreviature consisting of Capital Letters, signifieth, Rabbi Moses Ben Maimon, Rabbi Moses the son of Maimon. So Rabbi Levi, the son of Gersom, in his minority was called the son of Gersom, afterward Levi the son of Gersom at last רלבג, Ralbag, Rabbi Levi the son of Gersom. This distinction of Scholars, Companions & Rabbies, appeareth by that speech of an ancient Rabbi, saying I learned much of my Rabbies, or Masters, more of my companions, most of all of my Scholars."
Whatever one makes of Goodwin's interpretation of rabbinic literature (ספרי חז''ל)--certainly he is correct to reference the Arukh which cites R. Sherira Gaon (אדונינו שרירא ראש ישיבת גאון יעקב) on the different titles used by the חכמי התלמוד, the sages of the Talmud, I am most certain his scenario of How the Rambam Got His Name was wholly imaginary.
Modern historical research, of course, shows that rabbinic titles and how they are conferred vary from time to time and from place to place. For example, Moreinu was a European title no longer extant (so was haver).
 Here is the entry under the heading אביי Abbaye (from the 1553 edition of the Arukh printed in Venice by Alvise Bragadin):