When you die, rather than respawning, you lose your corporeal form. In your ghost form, you float about (think perpetual jetpack, however you move more slowly), partially translucent, and armed with a modified Insta rifle (the "ghost gun" -- grey-blue in color; still instant kill, but the delay between shots is much higher, and it's harder to use).
Living players can see you, but cannot harm you. To respawn, you must kill a living player. When that happens, you take over where they die and regain your corporeality, and they become a ghost. You only get frag points when you're alive.
Think "Revenge" in Bomberman. I've never seen this done in a first-person shooter, but I think it could work really well if it was done right.
Thoughts?