Ocean observations are expensive and difficult to collect. Designing effective ocean observing systems therefore warrants deliberate, quantitative strategies. We leverage adjoint modeling and Hessian uncertainty quantification (UQ) within the ECCO (Estimating the Circulation and Climate of the Ocean) framework to explore a new design strategy for ocean climate observing systems. Within this context, an observing system is optimal if it minimizes uncertainty in a set of investigator-defined quantities of interest (QoIs), such as oceanic transports or other key climate indices. We show that Hessian UQ unifies three design concepts. (1) An observing system reduces uncertainty in a target QoI most effectively when it is sensitive to the same dynamical controls as the QoI. The dynamical controls are exposed by the Hessian eigenvector patterns of the model-data misfit function. (2) Orthogonality of the Hessian eigenvectors rigorously accounts for redundancy between distinct members of the observing system. (3) The Hessian eigenvalues determine the overall effectiveness of the observing system, and are controlled by the sensitivity-to-noise ratio of the observational assets (analogous to the statistical signal-to-noise ratio). We illustrate Hessian UQ and its three underlying concepts in a North Atlantic case study. Sea surface temperature observations inform mainly local air-sea fluxes. In contrast, subsurface temperature observations reduce uncertainty over basin-wide scales, and can therefore inform transport QoIs at great distances. This research provides insight into the design of effective observing systems that maximally inform the target QoIs, while being complementary to the existing observational database.