What, if any, paleoclimate data can be derived from stromatolite fossils? Stromatolite fossils have a layered structure vaguely reminiscent of tree rings, which are a well-known source for climate data. Although the formation process of stromatolite layers is less seasonal than that of tree rings, it seems plausible that some information about environment conditions and changes in those conditions might be derived from stromatolite fossils. Having very close modern analogues in modern stromatolites might make interpreting such observations easier than for growth data from other fossils, and the great age of some stromatilite fossils might give at least a blurry window into such ancient environments.

Is there any potential for using stromatolite fossils to infer climatic (or other) conditions from billions of years ago?
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.