On Surnaming Dolls

Would you give your BJDs a surname?

Yes. Since the Main group of my dolls are "medievalish" fictional in genre, there are a few basic kinds of surnames typically used in that setting:
Descriptive, (Jon the Tall, Jon Hammerhand, etc)
Patronymic (Jon Larsson, Jon Mac Duncan, etc)
Locative: (Jon Bywood, Jon o' Kent, Jon o' Darbyshire)
And Occupational: (Jon the Hunter, Jon Tanner, Jon Taylor)

So Kevya is Kevya the Wolf, and that's a descriptive name. My harpist is Aeron Winter, another descriptive surname, even though his mother's surname was Pryderi. I have a knight named Sir Malcolm Owen, and it's a patronymic name, as his family is the House of Owen. The Prince/King is Lir, known to his people as Lir The Wanderer- I haven't decided on Lir's family name yet.

Would you give them YOUR surname?
Certainly not. My surname is for members of my family. My own surname is a very specific locative surname referring to a certain ancestral farmstead in a certain country. As there are less than two hundred of us by our name on this broad blue Earth, we bear it with a certain amount of pride, and use it to determine who, out in the wide world, is some distant kin of ours. Dolls aren't real people, and they are not in my family, nor are my dollies my "children", so they do not get my surname.

Would you rather give them a random surname?
Their surnames actually tend to mean something, as I mentioned above. Medievally, surnames were to tell people apart when they had the same first name, and also to impart extra information about them, such as what they were like or what they did for a living.