Why is the name "Phase vocoder"?
A vocoder is a type of synthesizer originally developed for speech (or the human voice, thus the "vo" prefix), even before the use of digital processors.  The Bell Labs Voder was demonstrated in 1939, and VoCoders were reportedly used for encrypted voice communication during WW2 (according to Wikipedia).

DFTs can be used to analyze which audio spectrum peaks are to be used to "code" the speech, and short STFT windows are used to get a fine enough time resolution to isolate typical phonic atoms over reasonably stationary time periods.

But the frequency bin centers of a short FFT are not spaced with sufficient frequency resolution to capture pitch, overtones, and modulations thereof, accurately enough to sound human. 

But, by taking into account the FFT phase information, the difference in the phase of FFT result bins between adjacent offset windows can be used to adjust (or interpolate) the apparent or actual frequency of the voice codec synthesizers to more accurately model narrow-band frequency peaks than by just using the magnitude alone of the STFT FFT results.  Thus the "phase" prefix to the analysis-resynthesis vocoder process.

Note that if you slide two Windows apart, the amount of phase change of stationary spectrum should be linear with frequency.  If analysis shows a little more phase change, the frequency is higher, a little less phase change, lower.  And this phase difference will be measurable even for differences in frequency much less than the Fs/N bin spacing of each FFT.  Thus there is information gain in using succesive frames and phase, rather than single frames and magnitude only, for the frequency estimation of narrow-band spectral peaks.  Phase can either be stretched or compressed across frames during FFT resynthesis, or the frequency of higher resolution oscillators can be adjusted to match the estimated frequency during synthesis.

With increases in DSP performance, phase vocoders are now used for music as well as speech, but the "vo" prefix remains.