The great thing about being a writer is, unless you are doing a non-fictional account of something, you can do pretty much whatever you want to do. Especially if you are including vampires and demons as antagonistic characters. Unless you are trying to get as close to reality as you can while including bits and pieces of fantasy, that is.

I'm not all up on Demonology or Vampire and Demon lore, but I'm fairly sure a pact that binds a demon to a human would have existed. Although I don't think saving a demons life would be part of the pact. I would try to swing a mutual agreement between the two, basing the Demons intentions on borderline parasitic reasoning. Both parties have something to gain in return for their working together, etc.

As far as the demonic name goes, what sort of personality are you going to give him? Are you using the levels of hells as described in something like Dante's Inferno or are you going for a general idea of a chaotic plane?

I've always found demonic names to be hard to pronounce and somewhat glutteral with little-to-no vowels in them. Something like Zzyxx if it would work. That isn't a very good suggestion for an actual name, just an example of something to build around. Knowing what kind of personality and history you are going to give the demon would help with suggesting a reasonable name.

EDIT: As an afterthought on the nature of the main character, since you used the term "slave" to describe her relationship with the demon I would figure that she has somewhat evil intentions or generally thinks bad about the character of the demon. As such, maybe just write it off as her not thinking he is important enough for a name, with insulting terms used instead. Or general "Demon" and other associated terms.