What methods are there currently that would lead to calculation of electron-muon mass ratio? The muon mass is approximately 207 times that of the electron mass.
Are we any closer in 2018 to calculate this value from first principles?
Are there any respected theories that get anywhere close?
If we are able to calculate it, it should be able to be calculated in terms of mathematical constants. (Or are there any cosmological constants that would affect the value of approximately 207?)
 A: In the standard model, the masses of elementary fermions come from yukawa couplings to the Higgs field, which are free parameters. 
In a grand unified theory, there may be some relations between yukawas, and in string theory, the yukawas should be completely determined by the specific choice of Calabi-Yau geometry, etc. 
So in these more advanced theories, there really should be exact algebraic expressions for quantities like m_mu/m_e. 
However, because of the energy dependence of such quantities in quantum field theory, they only assume a simple form at very high energies. At lower energies, there are numerous messy logarithmic, etc, corrections, due to "renormalization group flow". 
In the case of a grand unified theory, although there are fewer free parameters at high scales (fewer than in the standard model) because of the unification, there are still free parameters. So a GUT explanation of the observed m_mu/m_e ratio is just about fitting those GUT-scale parameters to the data. 
In principle, a string theory vacuum that reduces to the standard model at low energies, will provide exact predictions of the SM parameters. However, in practice that would require determining e.g. the size of the various extra dimensions in the lowest-energy state, and such things are still hard-to-impossible to calculate. 
So in both grand unified theories and in string theory, people don't try to explain e.g. that ratio of almost 207 that you cite. Instead, they just want to explain order-of-magnitude relationships among the particle masses. They'll say things like, there's a mass scale, and one particle's mass is about a tenth of that, and another particle's mass is about a hundredth of that, because the first particle's mass is suppressed by one (Feynman diagram) loop, and the second particle's mass is suppressed by two loops. 
There are a lot of papers like that. I suppose they are respectable if the calculations are correct and the hypothesis is not too contrived. But we are very far from getting the exact ratios right, and knowing that we are getting them right for the correct reason. 
I would stop writing here if I just wanted to give a cautious, sober answer that experts would endorse. The expert view is that the order-of-magnitude difference between muon and electron masses might have a simple explanation, but there absolutely shouldn't be a simple formula for the exact ratio, because of the messy effects of renormalization group running. 
However, there actually is a remarkable formula connecting the electron, muon, and tauon masses, the "Koide formula". It came out of a preon theory in the early 1980s, its success actually improved with time, and legitimate field theories have been constructed in which the yukawas are dynamical quantities that spontaneously satisfy this relationship. 
The main barrier to expert acceptance is that it is a high-precision relationship among low-energy quantities, and in fact, among quantities ("pole masses") that are defined at different energy scales. So I consider it remarkable that a mechanism was discovered which could actually preserve the simple relationship at low energies, by cancelling the corrections due to RG flow. 
This is described in Sumino 2008. The scalar potential which ultimately gives the desired yukawas is described on page 15, and the mechanism of cancellation on page 10. This is only an effective field theory rather than a fully renormalizable one, and there are some ad-hoc auxiliary hypotheses about coefficients, but in my opinion it deserves much more study. 
