You don't have to rhyme necessary as poetry in general is/can also be free verse (No rhyme or reason haha)
I'm arranging a track and I would like some lyrics for the part I added (the established always has them obviously); yet, maybe they aren't need.
I'd do the same thing lyrically as you would musically. Just rattle off stuff. I think where folks get in trouble in thinking too hard. Herb's in another thread regarding a user's track who's not sure where to go. (You know which, Herb)
I remember a tune for my sister's band where in the jam session, we just did random stuff. We did the "musical chain" (One person does this, the other does this and hopefully it connects, and so on). Once we'd laid the music down, my sis just sang about something silly. (I think the first line was: "It's 3am or something like that). Eventually, this turned into a real song with much better lyrics (not that we didn't put in the effort to come up with them, but we didn't think too hard.)
So, just go with whatever flows. If something doesn't work, don't scrap it. Save it for something else whether it be another part in the same song or a completely different one. (And don't be afriad to use repetition because it does help. Remember the Canon, Take Five, ... Exception examples. The latter two have lyrics, the former (at least to my knowledge) does not