The most likely candidates, at this time, are: British Forces Day (Monday closest to Armistice Day), April 23 (St George's Day as well as being Shakespeare's birthday), July 5 (anniversary of the birth of the NHS), and July 24 (Volunteers' Day, also touted as "Britishness Day").
More frivolous suggestions have included: Beatles Day on February 9 (the day the band first played at the Cavern Club) and Diana Day (on the Monday closest to Diana Spencer's birthday on July 1).
Recall that over the last 12 months, there have been many calls for a new UK public holiday, based on the fact that, at 9 public holidays, the UK are in second to last place in the EU, and the long gap between the summer bank holiday, in August, and Christmas (see our posts of December 26, August 28, July 24, and June 27). For that reason, the most likely (and also the one specifically mentioned by name by Gordon Brown) option would be to make the Monday after/nearest to November 11 a public holiday.