I thought it was the blue note but this is not it. For some reason, "perfect third" comes to mind, but google returns nothing. Anyone know?
In guitar tab it's labelled 1/4 tone bend. It has to be bent, as it's half way between the m3 and M3. Since it can't really be written on the normal staff of 5 lines, 4. spaces, in Western music it can't have a letter name, surely. The 'blue' third is generally regarded as m3 in a major key, and never heard of a 'neutral' third - or any other 'neutral' interval.
EDIT: having Wiki'd 'neutral third', it seems it is indeed a recognised term for that elusive note! Something new every day... Now what about the ones in between it and the m3 or M3..? Guitarists playing Blues use those as well !
When we talk about the 12 possible semitones, we are seeing music from a Western standpoint. Exploring other music traditions, we find other notes, and therefore other intervals, in common use. We think of them as in-between notes only because we are comparing them with our own standard set. The bent minor third, achieved on guitar by bending the string to increase the tension (and on harmonica by "bending" the major third down) is more common than the standard "western" minor third in Blues. As a reference, watch any blues guitarist, such as B B King, and you will see the minor third being played with the higher-sounding string visibly bent.
So as to what it is called, we could call it a bent minor third or, when it is clear we are talking about blues, perhaps we need only call it the minor third, it being taken for granted it will be played slightly sharp compared to "Western" intonation. On a guitar, if we play an A on the 4th string, 7th fret and at the same time a C on the third string 5th fret, and try various degrees of bend on the C note only, we can make a subjective judgement of the "relative sweetness" of the bent interval.
If we use the nomenclature of the 24 Equal Temperament Scale, the note midway between the major and minor third is called the Neutral Third. I am not aware of any frequency-measurement data comparing this with the bent minor third as played by recognized blues masters. Qualification: I have played guitar, including blues guitar, for almost 60 years, and have a degree in Applied Science.