There are a number of places to find interesting wood. If you don’t
know the location of a nearby hardwood lumberyard (not a regular
lumberyard that sells construction lumber), you can find hardwood
suppliers in the US and Canada using the Woodfinder website:
Most hardwood lumber suppliers expect you to come in and buy a board
(or boards) from their stacks. Most of their boards will be in the
range of 3" to 7" wide by 5’ to 8’ long, and around 3/4" thick (or
more). Many hardwood suppliers, however, have smaller cutoffs of
exotic wood and may even have pieces in their trash that are the
size you want.
You can buy lumber online from a number of places. Both of the
stores below will sell small amounts of wood in the thickness that
you mentioned (1/4"). These aren’t the cheapest places to get wood
that small, but they are big enough chains to be reliable. One or
both of these may have a local store in your area.
I think you may have trouble getting a smooth edge with a jewelers
saw (or a coping saw). If you cut the wood a hair oversize, you can
put down a sheet of sandpaper (maybe 120 grit) and smooth the edges
by rubbing the edge on the sandpaper.
You mention that your stone needs something warm around it. Cherry
has a nice warm color. It tends to have little flecks and stuff in
the grain (if you like that). Walnut is less red, more brown, and
probably has less going on in the grain. Your finish will have a lot
to do with the final look of the wood.
If you live near Chicago, I would be glad to cut a few pieces of
wood that are lying around to the size you want. I’m sure that there
are other woodworkers with pieces in their scrap pile that could be
cut to the size you want. If you can figure out how to find one of
them, they might cut a piece or two to size for you. If your local
community college has an art or furniture program, you might get what
you want through them.
The usual disclaimers apply: your mileage may vary, etc.