A key feature of many developmental systems is their ability to self-organize spatial patterns of functionally distinct cell fates. A spectacular example of this ability are artificial stem cell assemblies, which are paving the way towards a quantifiable self-organization of biological systems. However, while the relevant molecular processes are increasingly well understood, we lack conceptual theoretical frameworks for the dynamics and statistics of self-organized patterning. Specifically, it is unclear how to generically quantify the patterning performance of biological self-organizing systems, and how to identify the dynamical systems motifs that optimize this performance. Here, we develop an information-theoretic framework and use it to analyze a wide range of models of self-organization. Our approach can be used to define and measure the information content of observed patterns, to functionally assess the importance of various patterning mechanisms, and to predict optimal operating regimes and parameters for self-organizing systems. I demonstrate the application of our framework using experimental gene expression data of gastruloid and intestinal organoid symmetry breaking. This framework represents a unifying mathematical language to describe biological self-organization across diverse systems.