I have raynauds syndrome. Will building up the muscles and strength in my hands and fingers help?

No. Usually in raynaud's sx-- the peripheral blood vessels are very sensitive and vasospastic to decreases in ambient temperature. Often times the treatments are calcium channel blocker meds like Diltiazem and also wearing gloves and such to keep hands warm. Muscle exercises although healthy will probably not improve this tendency too much as this is an inherited trait. But exercise is always healthy.