On the site now occupied by the statue of King Charles was erected the original Queen Eleanor's Cross, a replica of whch stands in front of Charing Cross station. Mileages from London are measured from the site of the original cross.
When Queen Eleanor of Castile died in 1290, Edward I commissioned twelve crosses, each at one of the stopping places on her body's procession from Lincoln to Westminster. The original cross was replaced, then demolished (the stone being re-used used to make paving along Whitehall, round the corner), and in 1863 a rather ornate version (not strictly a replica) was put up in front of Charing Cross railway station, a couple of hundred yards away. Mileage distances on road signage are still measured from this point.
Nearby, at the South-East corner of the square, is Britain's smallest police station (says the London Encyclopedia), a phonebox-sized space inside the granite walls of a lamp post.
So how many other ways can the (exact) centre of London be defined?
Geography/latitude: Greenwich (meridian).
St Pauls/The Thames/Charing Cross as the centroid... or is it Camberwell, after London's expansion southwards? (and also: Hammersley has some interesting comments on geographic centres of continents)
Transport: Charing Cross station is the centre of London for Black Taxis. Piccadilly Circus is considered the centre of the Underground network (though Victoria is the busiest, and the first line ran from Farringdon to Paddington via King's Cross). London's Night bus routes all go via Trafalgar Square.
Roman London: The city's square mile roughly defines where Roman London stood, and there was a basilica and forum in Cornhill, dating from 70/90AD. Distances were perhaps measured in Roman London from the London Stone (if it were indeed London's "golden milestone"), which was placed in the walled city area. It sits today in a niche in a wall of a bank in Cannon Street.
In name: Apsley House - 'No 1, London', from which London's street numbering traditionally radiates.
People in the city...
The flow of people - commuting, shopping, socialising - multiple centres. Taxi drivers informally report that most of their journies merely enable other transport: they commonly connect railway stations to each other, or to Heathrow.
Retail and finance: where are the most/highest transactions and revenue? Oxford Street? City of London for non-retail.
Emotion and memory: where you were brought up. Your favourite place. The ideolocator: "You Are Here".