Besides GDP per capita, other common measures include life expectancy and refrigerators per 1000 people. Beyond that, there are measures of "quality of life" and "human development," which are intended to be somewhat broader (and, by implication, that much harder to measure consistently, especially on a global scale). The trouble with the more complex measures is that their definition is often politicized (how much you weight income vs. health care vs. environment vs. free time vs. etc.), and so they can be dicey. For example, the EU employs an entire army of beaurocrats who do nothing but invent measures that result in the EU comparing favorably to the US, despite the per capita GDP being so much lower.
Another issue is that most of the measures describe only averages, and so don't distinguish between countries with different levels of inequality. This is another factor that is often politicized; a preference for lower inequality over higher per capita income is pretty much the definition of socialism.
It may be, then, that there is no appropriate global measure of living standards in the broad sense. Every country can invent a measure that embodies their preferences and use it to claim that they have the world's highest standard of living, and they'll all be right (provided they allow people to migrate freely). So, there are two good reasons to use per-capita GDP: it's a number that can be directly measured and compared and, as per-capita GDP goes up, you can assume that the population in question has the resources to pursue policies that mesh with their priorities (regarding health care, free time, etc.). While it is problematic for making fine comparisons between similar countries, it has a good overall (large-scale) correlation with pretty much any reasonable measure you'd care to use.