You can name any card.
Be aware that Pithing Needle's ability excludes mana abilities. Activated abilities of the named card can still be activated if they're mana abilities.
A mana ability is any ability that (1) doesn't have targets, (2) isn't a loyalty ability, and (3) adds mana to a player's mana pool. If it's a triggered ability, it must (4) trigger from another mana ability.
Gemstone Mine's second ability is a mana ability (and its first ability is a static ability). Neither ability will be affected.
Fetches can be shut down, however, as their abilities are not mana abilities.
March 20, 2017 2:26 p.m.
Yes, you can name Gemstone Mine. As
Epochalyptik told already. I just wanted to correct one thing. If triggered ability triggers from triggered mana ability, it isn't mana ability. It has to trigger from activated mana ability.
For example if you have Caged Sun chosen green, Life and Limb, Conspiracy chosen saproling and tap Caged Sun for mana, only first trigger will be mana ability, but rest triggers are not mana abilities and can be responded to.