Have you looked at the work of Richard Garner and of Emily Riehl?
(Edit: having read Peter's answer, I've decide that I'm out of my depth when it comes to knowing which of them did what. So I'll just say "they" in what follows.)
I'm not an expert on this, but here's what I think I know. They have a notion of "algebraic" model category, which I think is rather more than having functorial factorizations. The idea, I believe, is that you know not just whether something is a fibration or cofibration, but why it is. (This builds on work of Marco Grandis and Walter Tholen on "natural weak factorization systems".)
That sounds like it's asking a lot, but they have a small object argument implying that any cofibrantly generated model category can be algebraicized. So, for example, this gives you a fibrant replacement monad (I mean a genuine monad, not just up-to-something), a cofibrant replacement comonad, and a distributive law of one over the other.
All I can find about this on Garner's website is this; I suspect he's done more, though. (Edit: Emily points out in her answer that his paper Understanding the small object argument is a better source.) The previous paragraph came from my notes from this talk by Riehl, and there's an associated paper of hers.