It usually goes for the dominant hand.
What do you do as a job, is it something repetitive with that hand?
Rest, rest and more rest and avoiding movement which may aggravate it is very important and if your job involves repetitive movements with that hand then you could look at ways of reducing that.
You could see an osteopath or chiropractor for further help, it costs but can be very worth it. Accupuncture could also help though get it done by a proper professional used to treating that kind of thing such as an osteo or chiro.
Yoga can help as well to regain strength.
Vitamin B6 can also help.
Some more info here from ARC...
http://www.arc.org.uk/arthinfo/patpubs/6008/60 08.asp