We can call it ProPhotoRGB because the monitor could reproduce every color within the ProPhoto RGB color space that represents a color that the human visual system can perceive. It would be useful because the monitor could display many colors that humans can see and print that are not displayable by monitors that have only 3 primaries. In some ways it is loosely analogous to printers which use more than 3 inks to achieve a wider gamut.
I carefully chose my words in the first sentence - "color that the human visiual system can perceive". The explation quoted from Hunt earlier in this thread explains it well, but let me try and rephrase and see if that helps. The eye can "see" or "detect" light in the electromagnetic spectrum between about 400nm and 700nm. The huge amount of information in this spectrum gets compressed down to only 3 numbers through our S (blue) cones, M (green) cones, and L (red) cones. To reproduce the color sensation of any arbitrary collection of light from the 400-700nm spectrum, we don't need to reproduce that whole spectrum (like in Lippmann Photography), but we do need to stimulate the SML cones the same way as the original light. If there existed 3 unique colors of light that could independantly stimulate the S,M, and L cones, we'd be there - these three lights (primaries) could be combined in different amounts to reproduce any color we can perceive. However, since the spectral responses of the S,M and L cones overlap, it is physically impossible to have a light (or a primary color in a display) that individually stimulates the S,M,L cones. Since we can't independantly stimulate the SML cones, we have to come up with some affordable number of lights (primaries), less than infinity, that can produce the same SML response as real world spectrals produce. If we pick 3 primaries that are close to the peak response of the S,M, and L cones, we can cover a good chunk of what we can perceive. That is the basis of NTSC, sRGB, AdobeRGB, and many other color spaces. But we can't cover everything we can perceive, as explained earlier. As we add primaries, we can cover more and more of the possible sensations, or perceivable colors, because we have more control over the response we want from the SML cones.
Here is the concept that takes some time to absorb: even though we can't *reproduce* any arbitrary color using only 3 real-world primary lights, we can still *describe* any perceivable color with only 3 numbers. This makes sense since we only have 3 types of cones. Describing all perceivable colors with only 3 numbers is what CIEXYZ does with the X,Y,Z axes. Two of the ProPhotoRGB primaries are imaginary colors - they do not represent colors we can perceive, but these numbers still serve a purpose - they allow us to describe more colors than color spaces based on perceivable primaries.
And this is why we can call it a ProPhoto RGB monitor.