Obesity prevents against osteoporosis, although it’s probably better to phrase it as ‘being underweight promotes osteoporosis’ (ref). (And, as an aside, despite an assertion to the contrary earlier in this thread, obesity increases the risk of osteoarthritis).
No one has yet mentioned the potentially beneficial effect of obesity on fertility. At first glance, that seems a paradoxical statement since obese women are often infertile (e.g. PCOS and allied disorders). The key point, though, is that weight loss, including that due to starvation, can restore obese women’s fertility. And that leads to an important advantage of obesity: preservation of ‘herd fertility’.
In other words, when normal-weight women are starving and losing weight (and are hence not ovulating and infertile), obese women who lose weight, though previously infertile, can cross a weight threshold where their fertility returns. So, during a period when the population is starving and most women are (temporarily) infertile, it is obese women who keep the ‘tribe’ going. This obviously has implications in terms of preserving and selecting for various genes involved in weight gain and weight loss, i.e. may help explain the high prevalence of obesity (or the tendency to become obese).