I can't comment on whether it is a commonly-attested name pre-Islamic sources (in my opinion, the sources are so scarce pre-Islam for Arabia it's unlikely we can form a firm opinion on this), but it's worth noting that a "Mu-" prefix is commonly used to denote a title in Arabic.

So a "Mu-Hadith" is a person who is a Hadith specialist, a Muhafiz is a person who is a custodian (typically of Qur'annic memorization), a Mu-slim is a person who submits, and a Mu-hammed is a person who is praised.

So literally it'd "one who is praised", rather than the less individualistic "praiseworthy".

This is definitely a theory that could hang together, as there's nothing in the Qur'an to allow us to decide that the few times the Qur'an refers to its prophet as "Mu-hammed" are either addressing him by name, or by title. So it's an unknown.

Unlike the Qur'an, the Hadith corpus of course does claim that Mi-hammed was his name rather than an appellate, but Hadiths are generally regarded by modern scholarship as post-hoc exegesis of the Qur'an rather than genuine (or at least taint free) historical memory transmitted parallel to the Qur'an.

The Hadith corpus also reports that Mu-hammad's father was named Abdullah ("Slave of Allah"), which is a rather incongruous name for an area supposedly suffused in "pagan" religious practices.

Similar to what /u/Atheist-Messiah says, it is hard to know for sure. I tried to answer this same question three months ago and this is what I found though. Although it existed in pre-Islamic times, its use was not common (at least among the Quraysh) and skyrocketed in popularity after Muhammad and Islam rose to fame and power:

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.

Wow, thank you guys for your thorough responses!