The naming of West End theatres is a pretty rum business. Even the most avid theatregoers are unlikely to be certain which Majesty it was that Her Majesty's Theatre is named after, likewise the Queen's. Likewise the Prince of Wales, likewise the Duke of York.

Theatregoers and non-theatregoers alike might wonder why the Comedy Theatre seldom has a comedy on stage. (At present there is a musical with an unhappy ending.) And whether the Fortune theatre was dedicated to good luck or someone's personal windfall escapes me too.

Over at the National Theatre, you'd have to be ignorant of the history of great acting and the history of the 20th century not to know immediately whom the Olivier theatre was named after. But equally you'd have to be an absolute expert on the minutiae of 20th century history to know anything about the two public figures whose names adorn the National's other two auditoria, the Lyttelton and the Cottesloe. And while I did once know why the Donmar was called the Donmar, I now, like nearly everyone who goes there, have forgotten.

The strangest thing about the names of London's theatres is that the people who make them what they are – actors, playwrights and directors – are in a small minority when it comes to the names outside the buildings. Laurence Olivier, of course, is remembered at the theatre where he was founding director. The Gielgud in Shaftesbury Avenue, formerly the Globe, was renamed after the great actor to avoid confusion with Shakespeare's Globe. Garrick, a great actor of a bygone age, is also remembered.

And that's pretty much it. Sir Cameron Mackintosh recently struck a blow for creative talent, renaming two of the theatres that he owns after Noël Coward and Ivor Novello. But in the main, fairly meaningless names such as Criterion and Lyceum and Palace abound, while actors and playwrights barely feature and directors don't feature at all.

I would certainly argue for Sir Peter Hall, a director who has been a colossus in theatre for 50 years, having a playhouse named after him. We could all choose an actor that deserves his own theatre. I would nominate Alec Guinness or Paul Scofield. But in this week of all weeks, it is a playwright who should be first choice when it next comes to renaming a West End theatre.

There was a slightly mischievous joke doing the rounds a few years ago that Pinter, when he had had a run of plays at the Comedy Theatre, mentioned to Tom Stoppard that perhaps the theatre might one day be called the Harold Pinter Theatre. Stoppard was alleged to have replied: "You might be better to change your name to Harold Comedy." It was a reasonable joke, but untrue and roundly denied by both men. What is true is that Pinter, one of the most prolific, thought-provoking and challenging playwrights that Britain has ever produced, must be commemorated by the art form to which he devoted his life.

I would suggest a playhouse that is better known for straight plays than musicals, perhaps the Duchess or the Fortune or the Criterion. But it doesn't really matter which playhouse it is. What does matter is that West End theatre owners decide which theatre will honour a playwright whose work may have divided people, may not always have been understood by everybody, but who enriched the stage, the English language, the understanding of human relationships and false and fractured memories, and pinpointed the disturbingly close relationship between comedy and menace. There needs to be a Harold Pinter Theatre in the city in which he spent his life.

Big name pulls crowd? No problem

I managed to catch the last performance of Rain Man in the West End. I was wary of going as I loved the movie, but the stage version proved a great evening. The producer Nica Burns showed me the results of a survey she had conducted among 1,000 members of the audience over a four-week period. Over 70 per cent of the audience was under 35; many had not been to the theatre before, and their main reason for coming was the presence of Hollywood star Josh Hartnett. So it's official – Hollywood stars do bring in a new audience. Thankfully, this one was a very fine actor.

Other findings in the survey were a little more sobering, at least for those of us who work in the media. More audience members, it turned out, learned about the show through word of mouth than from newspaper reviews and newspaper adverts put together. Maybe the people spreading the word of mouth had been reading the reviews.