Might be more work than the router option - but you could pilot drill a 7/8" dowel (or whatever dowel you can find that has a good fit in the existing hole), not much longer than you need to drill the 1-1/8" dia. hole, glue it into the stock, and then drill the 1-1/8" hole with a quality bit like a forstner bit. best to do with a good wood vice and a drill press.
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