The proposal that a female may benefit from choosing an attractive mate even if he provides nothing else, because her sons inherit the attractiveness and secure more matings. It is the sharpest version of Fisher's runaway argument and among the hardest predictions in evolutionary biology to test.
Female choice is well documented, as the sexual selection capsule sets out. What females gain from it is less clear.
Where a male provides food, territory or parental care, the benefit is direct and needs no special explanation.
The difficulty is species where males provide nothing but genetic material. Lekking birds, where males gather to display and females leave immediately after mating, are the extreme case. Some males obtain the great majority of matings, and choosiness is costly in time and predation risk. What is the return?

Ronald Fisher set out the logic in 1930 and Patrick Weatherhead and Raleigh Robertson gave it the name in 1979.

If a female's preference is heritable and the male trait is heritable, then choosing an ornamented male produces sons who carry the ornament and daughters who carry the preference.
Her sons are therefore attractive to the next generation of females, who share her daughters' preference, and they secure more matings than average. Her genetic contribution to subsequent generations is amplified through her sons' reproductive success.
The return is not better offspring survival. It is more grandchildren, obtained through her sons' mating success rather than through any quality the male passed on.
Fisher's further insight is that this is self-reinforcing. Preference and trait become genetically correlated, so selection on one drags the other, and the process accelerates until survival costs halt it. This is runaway selection, and it explains ornaments that appear extravagant beyond any signalling function.

The competing account holds that ornaments are honest indicators of quality, and that choosing an ornamented male yields offspring of both sexes that survive better.
The predictions differ testably.
Good genes predicts that offspring of preferred males survive better overall, in both sexes.
Sexy son predicts that sons of preferred males have higher mating success while offspring survival is unaffected or even reduced, since resources devoted to ornament are not devoted to viability.
The two are not exclusive and probably both operate, which is part of why separating them is hard.
Testing requires measuring the reproductive success of sons, in the wild, across generations, controlling for maternal investment. Very few studies manage all of that.
The best-cited support comes from work on great reed warblers by Dennis Hasselquist and colleagues, which found that sons of males with longer songs had higher reproductive success while daughters did not benefit, and that offspring survival was not improved. That is the sexy son pattern rather than the good genes pattern.
Studies of house finches and of some fish have reported similar results, and others have found the opposite.
A significant complication is differential maternal investment. Females mated to attractive males often invest more in those broods, laying larger eggs or provisioning more, which is a confound: the sons' advantage may come from their mother's investment rather than their inherited attractiveness. Experiments cross-fostering broods are the way to separate these and are rarely done.
Effect sizes are also a problem. Theoretical work finds the sexy son mechanism produces weaker selection than good genes under most parameter values, which means it may be real and secondary.
The mathematics is not in dispute. Fisher's runaway is a valid model, has been formalised repeatedly since, and produces the predicted dynamics under specifiable conditions.
The empirical question is how much of observed female choice it accounts for, and the answer is unresolved after ninety years.
Most current thinking treats the two accounts as ends of a continuum rather than alternatives. An ornament may begin as an honest signal of condition and become elaborated by runaway dynamics past the point where it indicates anything, which would make the same trait good genes in origin and sexy son in its later exaggeration.
It is filed as a hypothesis because the crucial measurement, that sons of preferred males gain enough mating success to repay the cost of choosiness while offspring quality does not improve, has been made in a handful of species and not established generally.