Is your cocktail a health food? That's a pretty tough sell. Ginger might be an interesting exception. It has been used as a digestive aid and and anti-nausea medicine for centuries. Studies have also shown anti-inflammatory, antioxidant, and anti-cancer properties in ovarian tumors (Rhode, 2007). Another recent study (Karna, 2012) showed that ginger extract destroyed prostate cancer cells in mice. However, the researchers still have a few things to learn. They extracted the ginger with methanol (gross!) and then evaporated off the toxic methanol for a dry extract. My vote (obviously) is to extract in ethanol and drink it!
Of course, the alcohol must be consumed in moderation or it has negative health effects. That said, the study would suggest that the presence of alcohol did not negate the positive effects of the ginger. The active compounds, such as 6-gingerol and 6-shogaol, are very soluble in both alcohols, but do not react with them much.
It is unlikely that we will ever mention health benefits on a ginger liqueur label. This is due to things like the $100,000 dollar fine paid by a winery for suggesting heart health benefits of red wine. Resveratrol in red wine does appear to have benefits, but label statements have to be justified at a very high level. So the bottom line is that, in my head, I will be thinking that that the benefits are real even if I can't say so.