Sand with #0000 steel wool, paying particular attention to going very lightly over the Weber logo you wish to preserve. The steel wool is really good at removing contaminates on the handles without removing much of the wood itself. Once it is cleaned, wipe with a damp towel and let dry in the sun. Seal and protect it by either spraying with Helmsman Spar Urethane or you can brush coat from a can of Helmsman.