We theoretically analyze the hybrid Mie-exciton optical modes arising from the strong coupling of excitons in organic dyes or transition-metal dichalcogenides with the Mie resonances of high-index dielectric nanoparticles. Detailed analytic calculations show that silicon core-exciton shell nanoparticles are characterized by a richness of optical modes, which can be tuned through nanoparticle dimensions to produce large anticrossings in the visible or near infrared, comparable to those obtained in plexcitonics. The complex magnetic-excitonic nature of these modes is understood through spectral decomposition into Mie-coefficient contributions, complemented by electric and magnetic near-field profiles. In the frequency range of interest, absorptive losses in silicon are sufficiently low to allow observation of several periods of Rabi oscillations in strongly coupled emitter-particle architectures, as confirmed here by discontinuous Galerkin time-domain calculations for the electromagnetic field beat patterns. These results suggest that Mie resonances in high-index dielectrics are promising alternatives for plasmons in strong-coupling applications in nanophotonics, with the coupling of magnetic and electric modes potentially providing additional freedom for external control, whilst offering new possibilities for tailoring artificial optical magnetism.