What, if any, paleoclimate data can be derived from stromatolite fossils?
Stromatolites have indeed been used in palaeoclimatic investigations. Here are a couple of papers I found through Google Scholar:

Paul I. Abell, Stanley M. Awramik, Robert H. Osborne, Sterling Tomellini, Plio-pleistocene lacustrine stromatolites from lake Turkana, Kenya: Morphology, stratigraphy and stable isotopes, Sedimentary Geology, Volume 32, Issues 1–2, May 1982, Pages 1-26, ISSN 0037-0738, http://dx.doi.org/10.1016/0037-0738(82)90011-2.

Andrews, J. E. and Brasier, A. T. (2005), Seasonal records of climatic change in annually laminated tufas: short review and future prospects. J. Quaternary Sci., 20: 411–421. doi: 10.1002/jqs.942 http://dx.doi.org/10.1002/jqs.942

As you'll notice, though, these are both relatively young records. I don't know if similar studies have been attempted for Precambrian stromatolites.