I spent around an hour looking but couldn't find too much. There is an article from 1936 (so methodologies have definitely changed since then) by Edward Jabra Jurji that claims that the name Muhammad has been in use in Arabia since the first-millennium B.C.E.

Jurji cites Kitāb al-Ishtiqāq written by Ibn Durayd (~10th century C.E) that the name Muhammad had been used prior to the birth of the prophet Muhammad in ~570C.E. However, it wasn't a customary name of the Quraysh tribe themselves. Muhammad's grandfather allegedly named him Muhammad so that he would "be praised in the heavens and the earth" (the name Muhammad is derived from the Arabic root ḥ-m-d meaning to praise and can literally be seen as the passive participle meaning "the one that is praised").

Just to be safe, I checked the Arabic of Kitāb al-Ishtiqāq and found the quote attributed to Muhammad's grandfather as well as a list of others who had been named Muhammad before the Prophet himself.

Jujri also said that the name Muhammad had been found in Southern Arabian inscriptions and cited the Corpus Inscriptionum Semiticarum (The Collection of Semitic Inscriptions). However, when I tried to find the references inside of it I could not locate them (Volume 2, Tome 1, #353 and Tome 2, #420). This is probably due to my own inexperience with the Corpus though.

So at least according to Jujri and Ibn Durayd, the name Muhammad had indeed at least existed before the prophet was born. However, at least among the Quraysh it does not appear to be very popular. This can be backed up by the fact that in lists of early notable Ṣaḥāba (companions of the prophet) there are only 1 or 2 other Muhammads found. However, after a few generations we can find 137 with the name Muhammad. As you can see on the second list, a lot of them are "Muhammad ibn xxx al-Ansari" meaning that they were the sons of the residents of Medina who converted to Islam when Muhammad made it his home after fleeing Mecca. Thus, much of the popularity of the name can be attributed to the prominence of the Prophet, and not necessarily that it was a popular name before.

Sources:

Jurji, E. J. (1936), PRE-ISLAMIC USE OF THE NAME MUHAMMAD. The Muslim World, 26: 389–391. doi:10.1111/j.1478-1913.1936.tb00898.x

Ibn Durayd, Abu Bakr Muhammad, kitāb al-ashtiqāq.