We study the origin and properties of extra or excess central light in the surface brightness profiles of gas-rich merger remnants. Combining a large set of hydrodynamical simulations with data on observed mergers (spanning a broad range of profiles at various masses and degrees of relaxation), we show how to robustly separate the physically meaningful extra light -- stellar populations formed in a compact central starburst during a gas-rich merger -- from the outer profile established by violent relaxation acting on stars already present in the progenitors prior to the final merger. This separation is sensitive to the profile treatment, and we demonstrate that certain fitting procedures can yield physically misleading results. We show that our method reliably recovers the younger starburst population, and examine how the properties of this component scale with mass, gas content, and other aspects of the progenitors. We consider the time evolution of profiles in different bands, and estimate biases introduced by observational studies at different times and wavelengths. We show that extra light is ubiquitous in observed and simulated gas-rich merger remnants, with sufficient mass (~3-30% of the stellar mass) to explain the discrepancy in the maximum phase-space densities of ellipticals and their progenitor spirals. The nature of this central component provides powerful new constraints on the formation histories of observed systems.
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