When 28-year-old Alexandria Ocasio-Cortez came from nowhere to defeat Rep. Joe Crowley in the Democratic primary for New York’s 14th Congressional District, she became a media darling overnight. Time magazine called her win “the biggest upset of the 2018 elections so far.” Within days Ms. Ocasio-Cortez was appearing on “The Late Show with Stephen Colbert” and “The View.”

Now Republicans looking for their own Ocasio-Cortez think they may have found her. Meet Elizabeth Heng—a Republican running in California’s 16th Congressional...