This paper was published in Phys Lett B in 2009, and predicted the Higgs mass to be 126 GeV based on the asymptotic safety of gravity. Is this prediction taken seriously by the theory community, or is it considered only to be a lucky guess?
EDIT: Thanks to Heidar for this fantastic compilation of Higgs mass predictions. This makes the point that it was probably a lucky guess, but nonetheless it could still be interpreted as (weak) evidence.
I would not say it was a lucky guess, it is simply a prediction of their assumptions. It is based on certain requirements for the value of the Higgs self coupling at the Planck scale which as far as I am aware translate into pretty much the same criteria that the Higgs potential remains just stable up to the Planck scale under renormalisation group effects.
Since the Higgs mass has been measured, there have been many papers which have discussed the fact that this mass lies just at the low end of stabilising the potential up to the Planck scale. However, what would appear to be the most accurate determination of this running including three loop pole matching effects etc is given here (they also include a discussion of the paper you cite). Their conclusion is that "while $\lambda$ at the Planck scale is remarkably close to zero, absolute stability of the Higgs potential is excluded at 98% C.L. for $M_h < 126$ GeV".
I understand this to mean that were Shaposhnikov and Wetterich to repeat their analysis using these more advanced calculations, they would predict a Higgs mass somewhat above 126 GeV.
This does not yet rule out their very interesting observation, and the Higgs mass measurement certainly lends weight to the asymptotic safety program (although somewhat diminished considering my above comments). Is the prediction taken seriously? Probably not that much, but only because the asymptotic safety program (maybe unfairly) does not get that much attention. It is however a growing field and the authors of the paper are certainly very well respected physicists.
Asymptotically safe gravity is extremely unlikely to work, as even Weinberg, it's creator, understands. It is pursued as an option, since you must exhaust all roads, but we know enough about gravity today to make it an extraordinarily implausible option.
The issue is the lack of holographic scaling in asymptotically safe gravity. It's an ordinary field theory, so it suffers from the near-horizon blow-up in black hole entropy which is common to all local theories of the formation and annihilation of black holes. This divergence was noticed by 't Hooft, as gave birth to the holograpic principle. The only known fix is to pass to an S-matrix theory, to string theory, in which case the formation and evaporation of black holes is consistent. This is the stringent criterion on quantum gravity, all the field theory conditions are far, far weaker.
This principle, the holographic principle, rules out all known approaches to quantum gravity except strings, and is the reason one can have confidence in the correctness of string theory without saying anything else. A one-parameter prediction is unlikely to be signficant evidence for asymptotically safe gravity, especially considering that it is theoretically nearly ruled out.
The analysis in the paper is also not particularly stringent in predicting $126~\text{GeV}$. It's a result of RG running with some ad-hoc assumption on the fixed point. They could have gotten essentially any answer less than $8~\text{TeV}$ by adjusting their assumptions, since they don't deal with the hypothetical strong gravity fixed point very much, beyond getting some guesses for the couplings from it. So this is just a lucky guess.
Asymptotically safe gravity is not really a viable option, only string theory is.
