I don’t have much on Mecklenburgh Square; it was named for Charlotte, queen of George III and Victoria Woolf lived there for a time. Oh, yes, George and Charlotte were patrons of Johan Zoffany, who gives his name to Zoffany Street.
Paternoster Square takes its name from Paternoster Row near St Paul’s Cathedral. It is argued by some that, before the Reformation, there was a regular procession of the clergy around the cathedral. This procession involved reciting the Lord’s Prayer in Paternoster Row, the Hail Mary in Ave Maria Lane, the Credo in Creed Lane, and the Amen in Amen Corner.( There are quite a few streets in London with religious (or religious-sounding) names, as you can see here.)
Another, less colourful but possibly more accurate, theory is that Paternoster Row, the oldest of the streets and dating from the 14th century, is where rosary beads (paternosters) were made. The other names may have followed on naturally in the religious context, especially as clerks who copied religious texts lived there.
Sloane Square is named for the physician, naturalist and collector Sir Hans Sloane. Among other achievements, he introduced cocoa to England and bequeathed the 70,000-plus objects in his collection to George II so that his collection would be preserved intact. That collection formed the basis of the British Museum.