You have a couple of basic choices: a clever name with a clarifying tagline or a basic name with a clever tagline. You want people to understand what you're offering without having to "dig" for it.
It sounds like you don't want a basic name, so that means that your tagline will need to clarify (for example, "In-Home Pet Services. Emergency Boarding Available."). Then, you can focus on you clever name, for example:
