I turned off the hot-water valve to my kitchen sink when I replaced part of the faucet. Now, the valve has a very slow leak at the base of the valve handle attachment. Is there anything I can do to repair this without replacing the valve?
Sure depending if this valve has a suffing box (Packing gland)
You could do the folowing
A- Open the valve fully and crack it closed slightly and this normally works If not GENTLY tighten the nut behind the handle until the drip stops.
B- Close the valve completly remove the handle GENTLY remove the nut and remove the old graphite packing and re pack with SPOOL teflon as graphite had been found to cause excessive wear and friction on the valve stem according to the NBBI and National board of US ship engineers and ASME.
Just make sure you do not loosen the bonnet nut connecting to the valve body as this would cause one heck of a flood.