Staying within the context of Vedanta it's the Jiva that is reborn, the individual soul, otherwise known as the mind or the ego.

Just like you said, it's who you think you are... so your associations/identity/desires/karmic impressions carry on to another incarnation, unless of course ones identity as the Atman is realized and the cycle is broken. It's not necessarily the end though, some realized souls continue to return to the world out of love, in service of the liberation of all beings.

It's right that the question is raised, as upon investigation there is no individual entity, no birth, and no death.