It could be a variant spelling of "Lambe" or "Lamb", which are both common Scottish surnames. These surnames likely originated from the occupational name for someone who raised lambs or sheep.
However, without more information about the specific individual with the surname "Lammie", it is impossible to pinpoint the exact origin.
It's important to note that surnames can change over time due to migration, spelling variations, and other factors. Therefore, it is always best to investigate the specific individual's family history for a more accurate origin.