In days gone by, when elections were often subject to irregular practices, voters could be bribed to cast their vote for a particular candidate by having drinks bought for them. Blue was the colour of the Whigs.
It was not unknown for wealthy C19th Whig landowners to insist on a change of name for pubs in villages they owned and to add the word ’blue’ to their name. People could drink “blue ale” in the “blue” pubs as an inducement to vote for their Whig landowner in the parliamentary elections.
I wonder if this was the case at Belmesthorpe? In the General Election of 1830, Rutland returned two MPs, Sir Gerard Noel, Bt. (Tory) and Sir Gilbert Heathcote, Bt (Whig.) Perhaps Sir Gilbert had a hand in the naming of the Blue Bell!