The name change will take place in February next year to pay tribute to the star's seven-decade career. Judi Dench co-founded the Theatre of Comedy, which staged plays at the Shaftesbury Theatre in the 1980s and 1990s.
She has also collaborated closely with the Taffner family and starred in the acclaimed BBC comedy series "As Time Goes By", which was produced by Taffner's production company.
Judi Dench says Shaftesbury has a special place in her heart, and theatre director Eleanor Lang says it is high time prominent women were given a greater place in West End theatres.
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