We present the H I emission project within the MIGHTEE survey, currently being carried out with the newly commissioned MeerKAT radio telescope. This is one of the first deep, blind, medium-wide interferometric surveys for neutral hydrogen (H I) ever undertaken, extending our knowledge of H I emission to z = 0.6. The science goals of this medium-deep, medium-wide survey are extensive, including the evolution of the neutral gas content of galaxies over the past 5 billion years. Simulations predict nearly 3000 galaxies over 0 < z < 0.4 will be detected directly in H I, with statistical detections extending to z = 0.6. The survey allows us to explore H I as a function of galaxy environment, with massive groups and galaxy clusters within the survey volume. Additionally, the area is large enough to contain as many as 50 local galaxies with H I mass < 10⁸ M_⊙, which allows us to study the low-mass galaxy population. The 20 deg² main survey area is centred on fields with exceptional multi-wavelength ancillary data, with photometry ranging from optical through far-infrared wavelengths, supplemented with multiple spectroscopic campaigns. We describe here the survey design and the key science goals. We also show first results from the Early Science observations, including kinematic modelling of individual sources, along with the redshift, H I, and stellar mass ranges of the sample to date.