I can't help much, but maybe this provides a starting point for someone else to continue.
I would update the Lidgren version, as it seems you're using one from almost two years ago.
Are you referencing a Lidgren assembly that is built with the same configuration (Debug/Release) as your game code? Have you made modifications to the Lidgren source code?
I assume you have not and am looking at the source code from the trunk of the Lidgren revision you apparently are using. The stack trace tells us the exception gets thrown by a line of code that looks like this:
m_socket = new Socket(AddressFamily.InterNetwork, SocketType.Dgram, ProtocolType.Udp);
MSDN documentation states that the constructor throws a SocketException when "the combination of addressFamily, socketType, and protocolType results in an invalid socket." InterNetwork, Dgram and Udp together should be a valid combination, not to mention the unlikelyhood such an error would not have been detected before a release. In other words, the docs are not much help here.
HResult -2147467259 is just "Unspecified error." so that doesn't tell us anything either.
Have you tried running the same code on another machine? Have you tried running it with administrator rights? I'm guessing this is something local to your machine, a firewall preventing the creation of sockets or some such. That line of code has no variables in it, the same constant values are used for everyone using it, and yet it's working fine for many users of Lidgren.