Apparently, there is a monastory not too far outside of the ski resort that people go and visit, but it will mean missing a days worth of skiing just as any sightseeing other than the village/town of Bansko itself will involve.
There are many places to visit in the region. Here are some of them worth according to me. Check on google for more info and location:
Rila Monastery
village Melnik and pyramids
village Kovachevitsa (a must)
town Sandasnki + mineral baths
village Delchevo (near Gotse Delchev)
Dobriniste and turkish baths (just 5 km from Bansko)