I'm no mechanic but I know a little bit about this issue simply because my 2006 Ram had the limited slip rear diff (or what Chrysler calls anti-spin) and I was extremely worried about this and did a lot of reading on it. I never experienced it but I also only kept the truck about a year.
I do believe that is a clutch pack retainer clip, commonly referred to as a c-clip. The third gen trucks equipped with the rear LSD are prone to these clips either breaking or simply coming loose and moving around inside the rear diff housing. This can cause two problems. First, it's obviously going to cause problems if it makes it's way into the gears. Some people open their diff cover to find the clips in pieces along with chunks out of the gears while some people simply find the clips laying in the bottom of the pumpkin. Second, my understanding is that those clips actually hold the gear assembly in the proper position. Once the clips are no longer in place to do this, the gear assembly can shift within the housing and actually begin to rub on the housing. Left long enough, it can damage both the gears and the housing to the point that the gears will eventually fail and the housing could possibly be damaged to the point of needing replacement as well.
I'm not 100% certain but the one you posted looks relatively intact. That would definitely be a good thing. Maybe that means it hasn't been in contact with your gears. Also, maybe it hasn't been out long enough to allow any damage to occur. The diff fluid would probably have that shiny particle look to it if damage had been occurring. I don't know how many are supposed to be in there but I'm pretty sure it's at least two. Are you certain there is nothing else loose in there? If not, maybe that one is the only one that has come loose. In short, maybe you caught this early.
I don't really know what the solution is. My common sense approach would be to have someone that knows what they are doing to look at everything and make sure it's in good shape and to replace the clips. There may be some other cure that will ensure that the problem doesn't duplicate but if there is I am not aware of it. I guess you could always swap to an OEM open rear diff as they do not have this issue but then of course you sacrifice the LSD.
Maybe someone else with a little more info about the cure will chime in. This is a pretty common, well-known problem with the third gen LSD trucks. In the mean time, just Google "Dodge Ram c clips" and you can read for hours.
**EDIT** I did a quick search and apparently swapping to an aftermarket gear assembly such as an Auburn or Detroit Truetrac is a solution as well. I didn't read it all because it's 20-some pages long but this looks like a good thread on this topic. Replace the asterisks with "dodgetalk".