Stefan Sakatiuk, 34, from Fauldhouse, West Lothian, was a passenger in the silver Vauxhall Signum when the car left the B7015 Sheephousehill Road and hit a tree.
The driver, who has not been named, left the scene of the crash which happened at around 7.45pm, 50m east of the Sheephousehill Road junction with Breichwater Place.
Stefan's father and two brothers said in a statement: "It is a terrible shock to the whole family. Stefan was a much loved son and brother, who especially loved his mum.
"We would like to be left alone to grieve and hope the driver does the right thing and hands himself in.
"Stefan will be dearly missed and we cannot believe that this has happened."
Lothian and Borders Police inspector Andy Amour said: "Inquiries are ongoing to trace the driver of the car and establish exactly what happened.
"We are keen to hear from anyone who may have seen a silver Vauxhall Signum on the roads around the B7015 before 7.45pm on the 1st of January.
"Additionally we would appeal for anyone who may know who was driving the car at that time to contact us on 0131 311 3131."