I don't have enough for a complete answer here (and I missed the bounty anyway) but I have seen some historical documents that might help narrow this down. To be clear: I don't claim any of the sources I cite here were revolutionary in their terminology (quite the contrary); they just happen to be things I was looking at recently that may be relevant.
Early 18th Century Italian term - tempo perfetto
There's some information which may be of relevance in Francesco Gasparini's thoroughbass manual, L'Armonico Pratico al Cimbalo (The Parctical Harmonist at the Harpsichord) of 1708 -- there's an English translation on IMSLP, which I recently started skimming through (like one does). Gasparini, an Italian composer who trained in Rome under Corelli and Pasquini, moved to Venice and became the director of music at the Pieta girls orphanage, where he is best-known for hiring a violin teacher named Antionio Vivaldi.
There's a translator's note which points out that twice, Gasparini uses the term "tempo perfetto" to refer to what we would call "common time". Note that this is contrary to the much earlier use of tempus perfectus to refer to triple time, as documented in @TomSerb's answer. In fact, the translator acknowledges this difference in terminology, and points out that the older sense of "perfect time" (triple time) hadn't been used for about two centuries by that point (emphasis mine):
On two occasions Gasparini speaks of "tempo perfetto" ("perfect
time"), but means thereby common time. In the first case (at Ex.31),
which deals with quarter-note motion, Gasparini's words "in tempo
perfetto, o binario" could mean "in perfect (triple) time, or in duple
time," or "in perfect, that is, duple time"; the quarter-note
progressions he describes, however, would be too complex for the fast
pace of \ the only triple time that could be involved. The second case
(at Ex.62) al- lows only the second interpretation, in which "tempo
perfetto" is taken as another name for duple, or common time. In any
case, the triple measures of the seventeenth century are—properly
speaking—not in "perfect time" (this not having been in common use
since the early sixteenth century), but are proportions, as Gasparini
elsewhere calls them.
So speculating heavily, it seems that somewhere between the beginning of the 16th and the beginning of the 18th century, duple time became known as "perfect" (or perhaps "complete"?) time, which could possibly be the origin of it becoming common time in English. One passage in which Gasparini uses this term is while discussing cadences. He is discussing what he calls a "greater compound cadence" and describes it as: "The greater cadence is formed in tempo perfetto, using four counts, in the following way." He then lists a shows of bass figures that essentially correspond to the progression V - I6/4 - Vsus4 - V7 all over scale degree 5 (before resolving to I). So maybe this tempo was "perfect" (i.e. "complete") in that it had enough counts in a bar to complete this entire cadential progression?
English term 'Common Time' is at least as old as mid-18th century
As far as the English side of things, I just happen to have also recently been looking at "A New Musical Grammar" published in London, 1746, by William Tan'sur (imslp link). In this book, the term "Common Time" is used as a synonym for all duple or binary time.
Common time is then broken into further "moods", indicated by various time signatures. Note that this usage of time signatures isn't quite the same as either today's usage, or as the mensural signatures mentioned by Tom Serb, and the modern system of notation.
This description of various "moods" of common time, although it does not match classical practice, matches very closely to the description of time in early American Shape Note Hymnals, such as one from 1800 listed on this page (search for the first occurrence of "common time" on that page).
So like I said, this doesn't completely answer the question of why, but hopefully it at least narrows down the search parameters a bit as to when.