I was warned by a friend that Mitad del Mundo was disappointing, so when my host family offered to bring me there, I accepted, but didn't get my hopes up. Therefore, I was quite pleasantly surprised.
Mitad del Mundo is one of those artificial villages that usually I would call a tourist trap, but it somehow managed to come off quite charming. All of the whitewashed buildings with adobe rooves are set up like a colonial village, ice cream shops abound, and on a sunny day, the atmosphere is just plain pleasant. Although we were there around noon on a Saturday, there weren't many people there, adding to the tranquility.
On weekends, there are live musicians and street performers. The day of my visit, there was a travelling circus, which was entertaining, though not earth shattering. That was followed by Ecuadorian performer who somehow managed to blend punchlines, slapstick, biting social commentary, and a plea to work for a better Ecuador into an amazingly funny stand-up bit. He's there most weekends, so if you understand Spanish, don't miss it.
There are a number of restaurants there. We had a delicious meal at El Karting, followed by helado on the terrace overlooking the plaza at one of Mitad's thousand ice cream shops.
There is an excursion from there to the nearby volcanic crater of Pululahua, which unfortunately I missed, but it comes very highly recommended.
The museums, I hear, are not particularly exciting, although the ethnographic museum inside the equatorial monument may be worth your time. Go to relax, take a picture on the Equator, have an ice cream, and enjoy the view.
The site is a short car/cab ride north of the city, or there is a bus you can catch along Avenida America at Av. Colón (walkable from the Mariscal district). I believe many of the northbound buses from the terminal pass Mitad as well. Entrance fees were about $1.