If the wire is rated at 350MHz, both stranded and solid will work the same (ie: no signal difference). You'd use stranded for the patch cords most likely as they're quite a bit more flexible and easier to maneuver. Solid is generally used for the long runs, probably because it's a bit cheaper, it's more readily avaialable and it also offers less attenuation to the signal passing through it but this would only be noticed for runs over 100'.
The RJ-45 ends are specific to stranded and solid cable, so make sure you have the proper end to match up with your cable.
I use solid Cat 6 throught my house, and even for the patch cords (I don't move around equipment around alot).