In fact, early Muslim scholars like Al-Mufaddal al-Dabi and Hammad Ar-Rawiya put a substantial amount of effort into compiling and preserving Arabic-language poetry from about the century before and during Muhammad's lifetime. Part of the goal was to preserve more of the state of the Arabic language at the time, to aid scholars in understanding the Qur'an; but, according to Islamic tradition, part of the goal was to revive knowledge of renowned literary traditions that had fallen out of favor. The movement was so successful, in fact, that a few scholars were accused of forging ancient poems just to appear to produce a collection of ancient works on their own. These poems, which generally survive in compilations from later centuries, are our major written sources for pre-Islamic Arabia.

Poetry in pre-Islamic Arabia was an oral tradition, an oral event. Poets gathered at prestigious courts like that of the Lakhmids to recite their poetry. The particular role of women poets in composing elegies to mourn the dead also became entertainment, as poets like Khansa (Tumadir bint 'Amr) competed in public competitions over the best elegy!

I can't promise a Meccan origin for any one poem. But it is worth pointing out that one collection, the Mu'allaqat, gets it name from the legend that the seven poems included were hung on or in the Kaaba. The much larger Mufaddaaliyat is noteworthy because it contains fragments as well as full poems, showing the scholars' (in this case Al-Mufaddal) dedication to preserving the pre-Islamic poetic tradition.

Jonathan Brown's 2003 essay on the social context of Jahili ("barbarian") poetry explores some of the scholarly treatments of the poems and has a bibliography I imagine would interest you immensely.