People are mixing terms. Many of the devices mentioned above (Wilson 3G, etc.) are for cell service to amplify their signal. They will not help at all for the type of 'WiFi' you seem to be asking about.
That is correct. It was unclear to me at first if the priority was voice or data. The Wilson amplifier would work for voice, independent of the neighbor, using over the air cell signals *if* the receiving phone has at least one bar to start with. But he specified data is the priority.
Are you able to try a desktop computer instead of a laptop? I ask this as I use several HP all-in-one computers and their wireless access receive capability is very good. I believe its because they put a better antenna inside due to the larger frame. This also explains why your laptop works better than your mobile-- antenna size. That is at least something to try that would not involve the neighbor. Also, there are USB network connections devices that include maybe 10 feet of cable and an external antenna. I've never used one but at under $50 those might be something to try. You could buy one of these, disable your internal laptop wifi, plug in the USB ethernet device (with external antenna) and try it instead. If you have a Fry's electronics nearby they probably have those.
Probably a sure winner is if you can add at least one more device to your neighbor system. If you could add an external outside "Access Point" (similar to a router but internally different), to the neighbor system, placed in an area with clear line of sight to you, that likely would solve your issue. Those are available for under $100 and it would need to plug in to his network via an Ethernet cable. This would likely make a huge difference as you would no longer be blocked by the walls of his structure, and any interference going on inside those walls (electric devices, ovens, etc.)
As stated above, the Nanostation products are excellent. I have no Ethernet cabling outside my ranch residence, so I use the Nanostation M5 from house to barn. Pointed at each other, I get excellent throughput so when in the barn, it seems like I am directly connected to the router. The throughput is so good I can even run video over it.
A Nanostation M5 would also require installation at the neighbor property, pointing at you. Unlike the Access Point, your laptop would not directly receive the signal-- a second Nanostation at your property would do that. You could then plug in your laptop to that Nanostation, or plug your own Access Point into it. I don't remember if the Nanostation offers broadcast wifi (such as to your laptop) or not. I only use my Nanostation point-to-point as a wireless bridge, then I plug Ethernet into each side of it.
Good luck with your investigation. These are always tricky to work out and usually require some tinkering before you get it right.