Well, your little friend can take care of itself where it is, you can still think of it as a pet, and maybe catch flies or roaches and put them in the web.
If you want to keep it in captivity, you migh read about keeping spiders.
It is not hard or expensive, but 'you' are responsible to see that it gets fed and has some form of water.
If this is a black widow, just respect it can have a bad bite and don't handle the spider.
http://boyslife.org/hobbies-projects/funstuff/1282/how-to-keep-spiders-as-pets/
http://www.wikihow.com/Keep-Spiders-As-Pets
Here is a house spider I keep in a deli tray:
http://www.flickr.com/photos/32083154@N02/8084201763/
And a black widow, they just need something to hide under:
http://www.flickr.com/photos/32083154@N02/3966394966/