Wouldn't 'kælenavn' do for 'specielle navn' ?
The sentence sounds a bit odd, anyway?!
I'm not sure i want to know her "special name"
Hendes specielle navn, is also correct, isn't it?
'Sit' refers to the subject in the sentence, which is 'Hun'(she). If you say hendes, it would imply the special name belongs to somebody else. So I suppose under certain circumstances it would work, but not something you would hear often.
Une phrase bien pourrie, ça c'est clair et net