A new bicycle rental system went into effect in July 2010. Operated by London Transport, information is available at this website; http://www.tfl.gov.uk/roadusers/cycling/11598.aspx
Here is the link to the map showing a good many ‘stations’ where you hope a good bike is available.
and here are the locations near Trafalgar Square where…
Charles II Street, @ Haymarket, SW1
Craven Street, WC2 @ Charing Cross
Embankment (Horse Guards), SW1
Little Newport Street, WC2, @ Charing Cross Road
Northumberland Avenue, close to Embankment, I think
Pall Mall East, @ Cockspur SW1
Panton Street, SW1, southwest of Leicester Square
St. Martin's Lane, WC2, @ Long Acre
St. Martin's Street, WC2, behind the National Gallery, (this might be your best bet).
Waterloo Place, SW1
Whitehall Place, @ Embankment
William IV Street, near the Strand