What exchange particles can change lepton flavor? So I know that in the standard model $W^+$ and $W^-$ can change lepton flavors but $Z^0$ cannot. I don't think photons can (is this correct?) and gluons can't because they would not interact with leptons. 
Can interaction with a Higgs Boson change a lepton's flavor?
If it makes any difference I'm interested in standard model behavior. 
 A: A charged lepton's flavor is determined by its mass. This is to say that an electron, muon and tau differ solely due to the fact that their masses are (roughly) $0.511\:\textrm{MeV}$, $106\:\textrm{MeV}$, and $1777\:\textrm{MeV}$, respectively.
In the Standard Model (SM), particles acquire their mass from Yukawa terms and the Higgs mechanism. After spontaneous symmetry breaking, the Yukawa terms give rise to both mass terms for the fermions and fermion-Higgs interaction terms. Some rotation of the fields is usually needed to read out the physics. In an appropriate basis, one identifies three massive charged leptons, whose interactions with the Higgs do not mix different mass eigenstates. Thus, interactions with the Higgs boson do not change the charged lepton's mass/flavor.
(The electromagnetic interaction, which arises in the kinetic term through the covariant derivative, is not broken by the Higgs mechanism and also does not change the charged lepton's mass/flavor).
For the neutrinos, however, the story is different. Neutrinos have been found to have nonzero masses. This right there is no longer "Standard Model behavior": in the SM neutrinos are forced to be massless by the symmetries of the model. If you insist in "Standard Model behavior", then the answer for neutrinos would be basically the same as for charged leptons. Nature did not choose it this way, however, so if you allow me to proceed...
Neutrinos have nonzero masses. There is also nonzero mixing, i.e. a massive neutrino is actually a linear combination of different neutrino flavors. (Conversely, each neutrino flavor is a linear combination of massive neutrino states).
Wait, how to define neutrino flavor, you ask? Neutrinos come in three different flavors: electron neutrino, muon neutrino, and tau neutrino. The flavors are defined by the neutrino's production method. If it was produced due the interaction of a $W$ boson and an electron, it is a electron neutrino (analogously for $\mu$ and $\tau$). Mind you, it won't keep being an electron neutrino for arbitrarily long times. Since the flavors are not aligned with masses, during propagation a neutrino oscillates. 
So if you want to change the flavor of a (neutral) lepton, simply let it propagate!
To go even further beyond the SM... If the origin of neutrino mass is the same as that of charged leptons -- the aforementioned Yukawa terms, at the cost of modifying the Standard Model field content -- one expects that interactions with the Higgs do indeed change neutrino flavor. However, keep in mind that the amplitudes for these interactions are proportional to neutrino masses, i.e. incredibly small. So you can forget about measuring them. (This is also true in see-saw models, although heavy extra-SM neutrinos are usually there and their coupling to the Higgs can be stronger).
