A Mrs Frances Livingston ran the Liverpool Hotel, Douglas in 1809 see http://www.ee.surrey.ac.uk/Contrib/manx/iomnhas/v061p122.htm
however the name goes back further
1752
Jane Livingston, wife of Archibald Livingston, of Castletown, died July 15th
see http://www.ee.surrey.ac.uk/Contrib/manx/manxsoc/msvol14/mem15.htm
1757 list of Castletown residents
LIVINGSTON Ken + Anna Livingston Maltster
http://www.ee.surrey.ac.uk/Contrib/manx/towns/castltwn/cstl1757.htm