The question posed came about during a 2nd Year Comp Science lecture while discussing the impossibility of generating numbers in a deterministic computational device.
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Randomness is not a binary property -- it's a value between 0 and 1 that describes how difficult it is to predict the next value in a stream.
Asking "how random can my values be if I base them on pings?" is actually asking "how random are pings?". You can estimate that by gathering a large enough set of data (1 mln pings for example) and mapping their distribution curve and behavior in time. If the distribution is flat and the behavior is difficult to predict, the data seems more random. The more bumpy distribution or predictable behavior suggest lower randomness.
You should also consider the sample resolution. I could imagine the results being rounded in some way to a milisecond, so with pings you could have integer values between 0 and 500. That's not a lot of resolution.
On the practical side, I would recommend against it, since pings can be predicted and manipulated, further reducing their randomness.
Generally, I suggest against "rolling your own" randomness generators, encryption methods and hashing algorithms. As fun as it seems, it's mostly a lot of very intimidating math.
As to how to build a really good entropy generator -- I think that's probably going to have to be a sealed box that outputs some sort of result of interactions on atomic or sub-atomic level. I mean, if you're using a source of entropy that the enemy can easily read too, he only needs to find out your algorythm. Any form of connection is a possible attack vector, so you should place the source of entropy as close to the service that consumes it as possible.