You could go to the town of Aserri just south of San Jose and ask about la Bruja Zarate and her precious stones.
Seriously, I may be mistaken, but I don't think CR is particularly known for precious stones in the same way that say Colombia is known for its emeralds or perhaps Guatemala is known for jade. Of course old pre-colombian jade and gold objects can be found in museums in SJ, but I don't think they'll let you take any with you.
More modern works of jewelry can be found in local shops but I don't believe the deals to be had in any of those are any more special than can be found elsewhere. I also don't think there are many if any at all newer sources of jade left in CR. However, gold is a somewhat different story. Strictly speaking gold is a precious METAL not stone but that may fit your bill. What exactly do you mean by "hunt for precious stones"? Do you just mean an outlet store where you can PURCHASE some stones or do you mean a place where you can try your hand at DIGGING some up?
There IS some gold mining in CR. Perhaps the most well known is in Juntas which is just off of the PAH on the "north route" up to Monteverde. I've never been there and don't think they have much more than a local museum let alone goldpanning or anything like that which would be open to the general public. Also I don't know how easy it would to get to by public bus. Probably the best way to visit would be by rental car or private driver on the way to/from Monteverde or perhaps other northern points in CR. For more info on this you could check out this link http://www.1costaricalink.com/elresidente/fa-11-12-2001.htm.
Another area that has some gold mining is near Carate (at the south end of Corcovado Nat'l Park on the Osa Peninsula), that is also mostly a local operation. In fact, the government even made payoffs a few years back to the local miners to get them to stop for environmental reasons but since then many of them have resumed their "wildcat" mining activities. Any way, if you don't mind shlepping yourself to this far corner of CR you could also try your hand at goldpanning too. Check out these links for starters http://www.lookout-inn.com/activities/gold_panning.php or http://www.osadiscoveries.com/adventure_tours.htm (towards the bottom of the page).
There are also more industrial (open pit) mining operations in various parts of CR but these have been subject of serious environmental problems and many of them have been highly curtailed or suspended such as the Crucitas operation up near the border with Nicaragua.