At Mitad del Mundo, or middle of the world, you can have the all-important photo taken of yourself with one foot in the Southern hemisphere and one foot in the Northern hemisphere. You can also buy a little something for the relatives back home: a genuine panama hat, for instance, or perhaps a stuffed llama.
Go on Sundays, that's when Ecuadorians go and there's traditional dancing and music. Also, be sure to visit the real equator line, before they built the tourist complex. It's a mini version.
There is now a fake line in Quito, about 10km away.