If the mineral spirits don't work out, two options would be to strip the piece, or to sand off the finish.
Stripping is always messy, but it would be easy going on a freshly applied finish. You would then have to re-sand for the new stain. I prefer the methylene chloride strippers. If you use one of them, read and follow the precautions on the label.
Going right to sanding would also work because stain doesn't really penetrate very far at all in the wood. However, you would have to allow the stain dry completely before sanding or you will gum up your sandpaper quickly. If you put the stain on really thick this may take a while and make the stripping option more appealing.
The size and shape of the piece of furniture would also be something to consider when deciding which of these would be easiest.