Have you looked at the work of [Richard Garner](http://www.comp.mq.edu.au/~rgarner/) and of [Emily Riehl](http://www.math.uchicago.edu/~eriehl/)?  

I'm not an expert on this, but here's what I think I know.  Garner has 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 Garner has 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](http://www.comp.mq.edu.au/~rgarner/CGNWFS/CGNWFS.pdf); I suspect he's done more, though.  The previous paragraph came from my notes from [this talk](http://www.math.uchicago.edu/~eriehl/ct2010slides.pdf) by Riehl, and there's an associated [paper of hers](http://arxiv.org/abs/0910.2733).