You can do it if you can get line of sight from your neighbor's house to wherever you put the antenna.
The solution Ken suggested is a good, simple way to go. A self contained weatherproof unit has the radio and the antenna together so there is no cable loss. You just run an ethernet cable from your existing router to the AP/antenna which is mounted in a good line of sight position.
Well, you have to get power to the box also.
There are other solutions using standard home routers with reflectors or with high gain antennas and perhaps cables. May or may not be cheaper depending.
Ideally, you get a nice $60 SOHO router/AP, put a homemade reflector on it, (see earlier link in this thread) and stick it in a window upstair with a view to your neighbor's house. Then you sit on the porch or by a facing window in their house and pick it up.
The router in the window solution only works if you have such a window.
Also, using a stronger antenna/rig on your laptop can help on the client end, if the reflector is not quite enough.
Forget the 10X range promotional BS. The keys to range are in antenna placement and gain. If you buy a home router, you want reliable G with a removable antenna so you can upgrade it if needed. Netgear might be fine, I don't know. The Buffalo WHR-HP-54G is the favored router this year. Not much publicity, but it's quite popular. BestBuy has em. Or newegg.
If you explain your existing home network including gear, locations and views to your neighbor, it will be easier to make specific suggestions.
Steve