Could a truly random number be generated using pings to pseudo-randomly selected IP addresses?

No.

A malicious machine on your network could use ARP spoofing (or a number of other techniques) to intercept your pings and reply to them after certain periods. They would then not only know what your random numbers are, but they would also control them.

Of course there's still the question of how deterministic your local network is, so it might not be as easy as all that in practice. But since you get no benefit from pinging random IPs on the internet, you might just as well draw entropy from ethernet traffic.

Drawing entropy from devices attached to your machine is a well-studied principle, and the pros and cons of various kinds of devices and methods of measuring can be e.g. stolen from the implementation of /dev/random.

[Edit: as a general principle, when working in the fundamentals of security (and the only practical needs for significant quantities of truly random data are security-related) you MUST assume that a fantastically well-resourced, determined attacker will do everything in their power to break your system.

For practical security, you can assume that nobody wants your PGP key that badly, and settle for a trade-off of security against cost. But when inventing algorithms and techniques, you need to give them the strongest security guarantees that they could ever possibly face. Since I can believe that someone, somewhere, might want someone else's private key badly enough to build this bit of kit to defeat your proposal, I can't accept it as an advance over current best practice. AFAIK /dev/random follows fairly close to best practice for generating truly random data on a cheap home PC]

[Another edit: it has suggested in comments that (1) it is true of any TRNG that the physical process could be influenced, and (2) that security concerns don't apply here anyway.

The answer to (1) is that it's possible on any real hardware to do so much better than ping response times, and gather more entropy faster, that this proposal is a non-solution. In CS terms, it is obvious that you can't generate random numbers on a deterministic machine, which is what provoked the question. But then in CS terms, a machine with an external input stream is non-deterministic by definition, so if we're talking about ping then we aren't talking about deterministic machines. So it makes sense to look at the real inputs that real machines have, and consider them as sources of randomness. No matter what your machine, raw ping times are not high on the list of sources available, so they can be ruled out before worrying about how good the better ones are. Assuming that a network is not subverted is a much bigger (and unnecessary) assumption than assuming that your own hardware is not subverted.

The answer to (2) is philosophical. If you don't mind your random numbers having the property that they can be chosen at whim instead of by chance, then this proposal is OK. But that's not what I understand by the term 'random'. Just because something is inconsistent doesn't mean it's necessarily random.

Finally, to address the implementation details of the proposal as requested: assuming you accept ping times as random, you still can't use the unprocessed ping times as RNG output. You don't know their probability distribution, and they certainly aren't uniformly distributed (which is normally what people want from an RNG).

So, you need to decide how many bits of entropy per ping you are willing to rely on. Entropy is a precisely-defined mathematical property of a random variable which can reasonably be considered a measure of how 'random' it actually is. In practice, you find a lower bound you're happy with. Then hash together a number of inputs, and convert that into a number of bits of output less than or equal to the total relied-upon entropy of the inputs. 'Total' doesn't necessarily mean sum: if the inputs are statistically independent then it is the sum, but this is unlikely to be the case for pings, so part of your entropy estimate will be to account for correlation. The sophisticated big sister of this hashing operation is called an 'entropy collector', and all good OSes have one.

If you're using the data to seed a PRNG, though, and the PRNG can use arbitrarily large seed input, then you don't have to hash because it will do that for you. You still have to estimate entropy if you want to know how 'random' your seed value was - you can use the best PRNG in the world, but its entropy is still limited by the entropy of the seed.]

Random numbers are too important to be left to chance.

Or external influence/manipulation.

Short answer

Using ping timing data by itself would not be truly random, but it can be used as a source of entropy which can then be used to generate truly random data.

Longer version

How random are ping times?

By itself, timing data from network operations (such as ping) would not be uniformly distributed. (And the idea of selecting random hosts is not practical - many will not respond at all, and the differences between hosts can be huge, with gaps between ranges of response time - think satellite connections).

However, while the timing will not be well distributed, there will be some level of randomness in the data. Or to put it another way, a level of information entropy is present. It is a fine idea to feed the timing data into a random number generator to seed it. So what level of entropy is present?

For network timing data of say around 50ms, measured to the nearest 0.1ms, with a spread of values of 2ms, you have about 20 values. Rounding down to the nearest power of 2 (16 = 2^4) you have 4 bits of entropy per timing value. If it is for any kind of secure application (such as generating cryptographic keys) then I would be conservative and say it was only 2 or 3 bits of entropy per reading. (Note that I've done a very rough estimate here, and ignored the possibility of attack).

How to generate truly random data

For true random numbers, you need to send the data into something designed along the lines of /dev/random that will collect the entropy, distributing it within a data store (using some kind of hash function, usually a secure one). At the same time, the entropy estimate is increased. So for a 128 bit AES key, 64 ping timings would be required before the entropy pool had enough entropy.

To be more robust, you could then add timing data from the keyboard and mouse usage, hard disk response times, motherboard sensor data (eg temperature), etc. It increases the rate of entropy collection and makes it hard for an attacker to monitor all sources of entropy. And indeed this is what is done with modern systems. The full list of MS Windows entropy sources is listed in the second comment of this post.

More reading

For discussion of the (computer security) attacks on random number generators, and the design of a cryptographically secure random number generator, you could do worse than read the yarrow paper by Bruce Schneier and John Kelsey. (Yarrow is used by BSD and Mac OS X systems).