Hann function is the original name, in honour of von Hann; however, the erroneous "Hanning" function is also heard of on occasion, derived from the paper in which it was named, where the term "hanning a signal" was used to mean applying the Hann window to it. The confusion arose from the similar Hamming function, named after Richard Hamming.
Harris, F. J. (1978). "On the use of windows for harmonic analysis with the discrete Fourier transform". Proceedings of the IEEE. 66: 51. doi:10.1109/PROC.1978.10837.
Blackman, R. B.; Tukey, J. W. (1958). "The Measurement of Power Spectra from the Point of View of Communications Engineering - Part I". Bell System Technical Journal. 37: 185. doi:10.1002/j.1538-7305.1958.tb03874.x.