Blackthorne definitely looks more finished.
You also find that in a lot of names they have "thorne" instead of "thorn"
Then again, you could use "Blackthorn" without the "e" to emphasize character quality - if the character is spiteful, mean and malicious like a "thorn". But if it is merely for a name then i suggest "Blackthorne" with the "e"
If I were you I'd think about your story, not about what readers prefer. What do you want your character to think of her surname? You could have an interesting little bit of characterisation if _she_ felt like it wasn't "finished" without the e, maybe she's tired of people spelling it wrong...