There are some constants other than the Higgs boson mass in the Standard Model that have been measured that can be combined in very simple formulas to give numbers that are within the margin of error of current Higgs boson mass estimates.
One is that experimental indications for the Higgs boson mass in the vicinity of 123-125 GeV are remarkably close to precisely one half of the Higgs field vaccum expectation value of 246 Gev. The other is that experimental indications for the Higgs boson mass are remarkably close to precisely half of the sum of the masses of the W+ boson, the W- boson and the Z boson (or alternatively, the sum of the masses of the W+ boson, the W- boson, the Z boson and the photon, since the photon mass is zero; or alternatively, the sum of the masses of all of the fundamental fermions, since the gluon rest mass is also zero). The sum of these masses is about 250.3 GeV, half of which would be about 125.15 GeV.
One could get an intermediate value by adding the sum of the relevant boson masses to the Higgs field vev and dividing by four (a result suggestive of a linear combination of the four electroweak bosons, the W+, W-, Z and photon).
Another way to bridge the gap would be to use the "pole masses" of the W and Z bosons, rather than their conventional directly measured masses. This basically adjusts the masses of unstable fundamental particles downward by a factor related to their propensity to decay in an amount proportional to half of their decay width, which at a leading order approximation is about 1.8 for these bosons. This would give us a sum of the three pole masses which is equal to roughly 248 GeV, which would be a fit to a 124 GeV Higgs boson pole mass if there is such a simple relationship, although the match would presumably have to be to the pole mass of the Higgs boson (something not yet possible to estimate with any meaningful accuracy as we have considerable uncertainty in both the Higgs boson mass and its decay width).
These pole mass calculations are approximate, and an exact calculation has significant terms at two loop level and probably beyond, so coming up with an exact pole mass calculation figure is a bear of a calculation. But, the notion that the Higgs field vacuum expectation value might be equal to the double the Higgs boson pole mass, and to exactly the W+ boson pole mass, the W- boson pole mass, the Z boson pole mass, and the possibly the (zero) masses of the photon and/or the eight gluons, is an attractive one. It is also suggestive of the idea that the Higgs boson itself might be understood as a linear combination of four spin one electroweak bosons to get a Higgs boson, whose pairs of opposite sign spins combine to produce an aggregate combined spin of zero, in line with the scalar character of the Higgs boson. One would need some reason to come up with a factor of two, or alternatively, some reason to add in the Higgs vev to the sum of the four electroweak boson masses which would naturally be divided by four since it is derived from a linear combination of four bosons.
The Z boson mass is related exactly to the W boson mass by the Weinberg angle aka weak mixing angle (an angle, for which the sin squared is about 0.25, which Rivero has some interesting numerological speculations in a 2005 paper and whose possible origins are discussed in terms of a heuristic set forth in a 1999 article). So, if there is some simple relationship between the Higgs boson mass, the Higgs field vacuum expectation value, the W boson mass and the Z boson mass, such that the Higgs field vacuum expectation value and Higgs boson mass can be calculated from the W boson and Z boson masses, it ought to be possible to derive all of these constants exactly, in principle at least, from the W boson mass and the Weinberg angle (which is definitionally derived from the relationship between the weak and electromagnetic gauge couplings).
Of course, even if the experimental values come to be known with sufficient precision to rule out such simple relationships, one is still left with the question: why should this simple relationship be such a near miss? For example, does the simple relationship capture the first order terms of an exact relationship whose next to leading order and beyond terms are unknown?
Afterthoughts
This all becomes even more exciting if one can come up with a generalization of the Koide formula to account for all of the charged fermion masses from just a couple of constants. Both the W boson mass and Z boson mass were predicted from other constants known at the time before they were discovered, and one could conceivably get the number of free constants in the Standard Model down to a much smaller number.
While not precisely on point, this is also as good a point as any to ask, why have string theory and SUSY so utterly failed to provide accurate predictions of the mass constants or mixing matrix values in the Standard Model? Isn't that the very least that we should expect of any purported grand unification or theory of everything scheme?
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