Thomas Nast likely combined two elements. One is mitre worn by bishops that ends in a point. Saint Nicholas probably didn't wear it but in Dutch and German speaking areas popular depiction of him in a red bishop outfit (is there a better name for it? I am not familiar with Catholic vestments) has been around since at least the 16th century.

Another one is soft pointy nightcap commonly worn at night to keep warm (similar to the one worn by Mr. Scooge).

Note that almost all of Nast's illustration of Santa feature a suit made entirely of fur instead of felt or velvet with only fur trims. This came from the poem A Visit from Saint Nicholas where he was described as "dressed all in fur, from his head to his foot".