The usual answer is blast them with more light and more flow. Which IMO is correct with some caveats. First, you really can't have too much flow in an SPS tank. Corals need flow to breath, They can't swim closer to the surface nor do the have muscles to force water around them to move so you have to do it for them. Second, lighting. This is a hugely debated topic among reef keepers for a variety of reasons. The things we tend to agree on is the more PAR the better, Kelvin seems to effect our eyes more than the coloration of a coral as the same coral under a 5500k light is could look more brown than the exact same coral under a 20k light (more on this later). Lastly keeping the water at favorable parameters has a lot to do with it. Alk and Calc allow the coral to grow and sustain its life while pH is important for calcification. The exact relationship between these parameters isn't as important but keeping them inline and IMO more importantly consistent does seem to have a huge effect on corals themselves. As side note to this is keeping your water clear. The clearer your water, the better the PAR you are going to get as all those little particles floating in a tank do cause a drop in PAR and sometimes substantial. This also goes for cleaning the glass and bulbs frequently. You must be careful when cleaning the bulbs, but remove salt creep and the like from anything between the bulb and the water surface.
OK now that my basics are over with, now comes the controversial part and some I would love to discuss/argue with people just to get their thoughts.
Feeding. Corals get most of their energy from light. This allows them to grow, color, etc. However, additional energy comes from the food they intake. This gives them more energy for their to process. Some people target feed zooplankton, some blast the tank for hours with a coctail of food and others simple feed the heck out of their fish and let the poop do the talking. All seem to work. Which one works best is debatable and most likely depends on the individual coral.
Nitrates. This is a balancing act. Corals are really plants and they like to eat. Nitrates indicate how many organics are in the water column. So it makes sense to some degree that a small amount of nitrates is good for corals. In my experience and reading from other reefers, increasing the nitrates seems to actually work well for corals. That said higher nitrates causes other problems including annoying algae. IMO the best scenario is to run the tank with low nitrates for a day or two and then zero the rest of the week. However implementing that is difficult to say the least.
Kelvin. This is hot topic. To me, Kelvin firstly changes the way you see a coral. The more you go towards the 20k mark the bluer the bulb and the more some colors pop. I find the more you move toward the 20k bulb the more purple some corals appear. This is more your eyes than the corals coloration chaning. The problem with moving toward the 20k away from 5500k is that it looks great to you, but for energy purposes the coral gets more usable energy from 5500k. So the closer to 5500k you get the faster the corals tend to grow as the PAR rises. However, this leads me to probably one of the more controversial items... growth.
Speed of growth tends to effect a corals color. From reading and watching my own tank, the faster a coral grows the more brown it tends to become. Before you hit the reply button and complain about that statement hear me out. Good growth often is an indicator that all the params in the tank are spot on and in most tanks growth and coloration go hand in hand. I believe many reefers just assume that since they have good color and good growth that the two are linked in a positive way only. I think that corals that are growing very fast tend to actually loose color as it requires less energy to maintain the brown color than any other color. The faster a coral grows the more energy it needs for that growth. So the main body tends to become more dull as the tips grow faster. This is why I think when people shift to 20k bulbs away from lower kelvin bulbs not only do their eyes see a different color, but the color of the coral changes as well over a period of time since it has less PAR it is growing at a slower rate which I believe enhances the overall color. This is all IMO and I am happy to admit I could be very wrong here so let the SPS experts chime in.
UV is one of the bigger causes of coloration in the wild. Simply put, coloration is often sun screen for a coral. It shifts color to protect itself from too much harmful radiation. Hit some corals with more UV and you will find they will color up nicely. Actinic which a lot of people here will tell you is strictly for you eyes and doesn't effect the corals is possibly wrong. I think some powerful actinic lighting actually causes corals to flouresce more. MH lighting often provides this function, but hitting your corals with more of this dedicated spectrum can cause certain corals to almost glow even after the lights get turned off.
Food/protiens are the hot new way to color up your corals... or at least they will be soon enough. The interaction of proteins, corals and lighting is where a lot of researchers are beginning to focus their attention. It seems that some proteins actually shift the color of a coral. So some SPS farmers and researches are trying to develop compounds and general food products to unlock this part of coral coloration. I have a ton of reading to do on the subject still so I won't comment on the specifics, but watch for commercial and other endeavors in the short term future that are basically an additive that colors your corals.
Dyes... yep dying your coral. This seems strange, but there is some evidence that the coral skeleton when it calcifies can be shifted away from white. If true and it works we may find corals that are far more brilliant. So far I have only heard of some SPS farmers working it out, but I have been told that it is almost perfected by some. Again, don't know the specifics but find it fascinating.
More to follow a bit later...
Ultimately each coral seems to have its own wants and needs. On a fundamental level they are all the same, but once you move past the basics corals tend to color up better in different circumstances. One green slimer in my tank can simply react differently than another green slimer from a different mother colony in the same tank. However, generally speaking if you hit them with good water params, lots of light and plenty of flow they will respond with growth and color.
