Can a TLS client perform any DH regeneration? I have only the barest understanding of cryptography, but I've been following the Logjam issue; some posts and articles (elsewhere) imply that a client can regenerate DH keys. Is Logjam solely a server issue? Is there a way for the client to perform DH regeneration on a regular basis; if so, how? Is there any way the client can circumvent the Diffie-Hellman Logjam issue?
A client can indeed "regenerate" DH-keys.
But this simply means he re-connects to the server. The Logjam attack works by recording the connection establishment and then compute the derived private key. For this attack to work the domain parameters of the DH-exchange need to remain unchanged as precomputation is neccessary. If you just reconnect the server will present you with the exact same group and the attack will only take seconds to decrypt the session key. For Logjam to be defeated one needs new domain parameters, which take a while to be generated, hence this isn't done on every connection, usually this isn't even done in years.

Logjam isn't solely a server issue. If the vulnerability is fixed on either the browser (how-to: next paragraph) or server-side it's closed. Both sides need to support it and all major browser companies (Mozilla, Google,...) already announced patches to close this vulnerability.

The client can circumvent the issue by requiring 2048-bit+ primes for the domain parameters of DH, if the client decides to use DH-exchange.
The best way to avoid Logjam is to avoid DH, by choosing ECDH, which isn't vulnerable.This is also the way recommend by the authors of the accomplying paper.