Visitor Yama says:
"I have had *very* nice results using plain ol' blueberries and other berries. Basically the reasoning goes as this: anything that will stain that white tablecloth you cannot use until the last kid is off to college, will probably/likely stain wood.
YMMV, etc. and likely light will degrade such color, but so far my blueberry puzzle is holding after 6 months."
Once again I can't get a clear answer as to whether these products are truly safe for toys, and they don't come in a huge range of colors, but I'm including them here for the sake of completeness.
Recommended for outdoor use only! - these stains contain a solvent that dries to a non-toxic finish, but is not appropriate for indoor or small toys.
Barbara Butler makes play structures and stains with her own Tung Oil Wood Stain colors. I'd definitely like to give these a try and may see how one of her small palette sets works out - it's not cheap, but it's not as expensive as several of the other options described here. It is described as non-toxic and used on equipment for children, but I have not contacted them to ask if their stains would be acceptable for use on smaller toys that might be chewed on. That said, their stain seems to be based on the now-defunct Woodburst Tung Oil stains.
Not recommended by the manufacturer for toy use. I contacted Minwax to inquire about using their stains on toys, and sadly they told me that they cannot recommend their products for such a use (despite several posts in their forums claiming that it would be just fine to do so.) This is not a huge surprise given the broad base of applications for Minwax.