Well, let me at least make one correction to what you've laid out:
If you've given them access to your wireless access point, they are on your home network. What they can see depends on a couple of things, but to be blunt: I hope you trust them.
It's good that your wireless access point is using WEP encryption. That helps prevent others from accessing your network. In giving your neighbor your WEP key, though, you've given them access to your network. All of it. It's pretty much the same as giving them a cable directly in to your router.
That means that, for example, if you share files or your printer, your neighbor can see them, and possibly access them.
This is the part where you need to trust them. Allowing them on your home network could allow them access to any private files you might have on your computer.
Whether or not they can see your surfing habits depends on your router. If it is indeed a router (and it sounds like it is), chances are their computer cannot see the traffic to and from your computer. A router is typically a smart device that "routes" the data only between the ports that are actually involved in the conversation. However that's an assumption we make of the device, and since the true definition of what a router should or should not do is somewhat vague, it might not always be the case.
Finally, the conversation between you and your bank should be happening over a secure "https" connection, and this is a great example why. A secure connection encrypts the data so that it cannot be seen by anyone other than the two participants in the conversation: you and your bank. This is a computer-to-computer encrypted conversation, so even if your neighbor could monitor the conversation, they wouldn't be able to understand it.
So in your shoes I would actually not be concerned about their ability to monitor your web surfing habits, but instead, I'd be concerned about their ability to see files on the computers in your home network.