You can't really see a lunar rainbow with the naked eye. Our eyes don't render color very well in the dark, so at best you can see a bright arc and maybe a hint of color. But the color is there, and film or digital sensors capture it well. One of the reasons that more people haven't photographed lunar rainbows from Cook's Meadow in the past is that you can't really see it from there. With film you'd just have to guess about the position and wait to find out whether you were right. With digital sensors it's easy to just take a photo and see the position of the rainbow.
Last night the moon was bright enough that I could see the rainbow fairly well. Along with everyone else I watched it move down the fall as the moon rose. When the rainbow fell too low, most people packed up (it was 11:30 p.m. after all), but I followed the moon westward and saw an even more vivid rainbow from near the Yosemite Chapel. This photograph was made from that spot. Rainbows form a circle around a spot opposite the light source—sun or moon. Lunar rainbows can be seen on any waterfall that gets moonlight if you position yourself properly. In fact I photographed a lunar rainbow on Castle Geyser in Yellowstone many years ago. Maybe I'll put that image in another post.
Long exposures with digital cameras can create lots of noise, but newer models handle this pretty well. You're usually better off making a long exposure at a low ISO than a shorter exposure with a higher ISO. But with lunar rainbows you don't want to make the exposure too long or the bands of color will blend together. I used a six-minute exposure at f/4 and 100 ISO for the image above. Trying to lighten a dark image will exaggerate noise, so it's best to make the photograph as light as possible without overexposing it. Don't rely on just how it looks on your LCD screen. Since it's dark, the screen will look bright even if the image is underexposed. Use your histogram.
Having said that, a nighttime photograph shouldn't be too bright. You don't want it to look like it was taken under midday sunlight. I actually darkened this image—especially the sky—in software to give it more of a nighttime feeling.