The location of your router and the neighbors computer are crucial, do you have line of sight with the neighbors machine from the router?
You may want to consider getting a linksys router and then installing modified firmware so that you can increase the transmission power. Or look into purchasing a wireless router with the N standard. If you get a router with the N standard, your neighbor will have to get an N stand card capable of using the signal to the fullest, I recommend the following if you decide to take that route:
Router:
http://www.newegg.com/Product/Produc...82E16833127077
Networkcard for neighbors machine:
http://www.newegg.com/Product/Produc...82E16833127213
Its best if you have the router and the machine have some kind of a line of sight, trees and tall fences will effect the signal somewhat. So adjust accordingly.
Good luck and I hope that this was helpful. Feel free to ask most questions if need be.