There is really only one solution to your problem. I do note however that you're not interested in upgrading your router, so I will talk a little about that.

Remember that by not upgrading your router, you are only delaying the inevitable.

What will work long enough for you to get a new router:

- PULL THE PLUG

This is honestly the best solution untill you get a new router.

- or TURN OFF THE WIFI

You can probably do this from the administration panel of your router, if not, remove the antennas from the physical device and go ethernet only. If that is not possible, unplug.

- or CHANGE YOUR WEP PASSWORD and TURN ON MAC FILTERING

If your router supports it, you can define a MAC address filter to either whitelist or blacklist devices accessing the accesspoint. To do this, you will have to log into the administration panel of your router and enable MAC address filtering. (Check your manual on how to do this). What kind of filter you can enable depends on your router. If it's a whitelist, you will have to find the MAC addresses of all your devices, and add them to the list. This way only the devices added to the list will be able to connect to the internet. If it's a blacklist, you will have to find the MAC address of your neighbour, and add that to the blacklist. This way, all other MAC addresses are allowed access, but not the addresses in the blacklist.

Why this won't work in the long run:

MAC filtering is fairly simple for someone to who's a bit tech savvy to avoid. There are multiple tutorials on the subject, and it's really as simple as running a few terminal commands. Since your neigbour has already cracked your WEP-key, chance is, he has the skills enough to google how to spoof a MAC address.

This way, the neighbour can "pretend" to be you, by authenticating with the router using your computers MAC address.

Changing the password won't last for long, because of how WEP works. An attacker only need to listen to your network for a short period of time, to be able to extract your password. You can read more about it on Wikipedia.

And at last, since you've been breached, anything you have connected to your access point could in theory be infected now. Not only could the hacker have changed your DNS settings, started logging network activity etc, he could also now be in the possession of all your files, pictures, passwords, credit-card information etcetera. To be honest, all of your devices (if not already hacked) are in great danger of getting breached any second.

What you have to do:

Get a new router. Seriously. WEP was deprecated in 2004 and has been deemed insecure for a long long time. As you noted yourself, there are a lot of information on how to crack WEP keys online, and it's (almost) as simple as running a terminal command. Anyone can do it. Remember that in most countries, you are liable for what passes through your network. So if an adversary downloads torrents, children pornography or bomb schematics, you will be held liable. Is prison better than spending 20$ on a new router?

[Edit 07.07.15 13:18] I have rephrased the answer to more correctly state the fact that a new router is the only long term option, as noted by @JamesRyan

There is really only one solution to your problem. I do note however that you're not interested in upgrading your router, so I will talk a little about that.

Remember that by not upgrading your router, you are only delaying the inevitable.

What will work long enough for you to get a new router:

PULL THE PLUG

This is honestly the best solution until you get a new router.

Or TURN OFF THE WI-FI

You can probably do this from the administration panel of your router, if not, remove the antennas from the physical device and go Ethernet-only. If that is not possible, unplug.

Or CHANGE YOUR WEP PASSWORD and TURN ON MAC ADDRESS FILTERING

If your router supports it, you can define a MAC address filter to either whitelist or blacklist devices accessing the accesspoint. To do this, you will have to log into the administration panel of your router and enable MAC address filtering. (Check your manual on how to do this). What kind of filter you can enable depends on your router. If it's a whitelist, you will have to find the MAC addresses of all your devices, and add them to the list. This way only the devices added to the list will be able to connect to the Internet. If it's a blacklist, you will have to find the MAC address of your neighbour, and add that to the blacklist. This way, all other MAC addresses are allowed access, but not the addresses in the blacklist.

Why this won't work in the long run:

MAC address filtering is fairly simple for someone to who's a bit tech savvy to avoid. There are multiple tutorials on the subject, and it's really as simple as running a few terminal commands. Since your neigbour has already cracked your WEP key, chance is, he has enough skills to google how to spoof a MAC address.

This way, the neighbour can "pretend" to be you, by authenticating with the router using your computers MAC address.

Changing the password won't last for long, because of how WEP works. An attacker only need to listen to your network for a short period of time, to be able to extract your password. You can read more about it on Wikipedia.

And at last, since you've been breached, anything you have connected to your access point could in theory be infected now. Not only could the hacker have changed your DNS settings, started logging network activity, etc., he could also now be in the possession of all your files, pictures, passwords, credit-card information, etc. To be honest, all of your devices (if not already hacked) are in great danger of getting breached any second.

What you have to do:

Get a new router. Seriously. WEP was deprecated in 2004 and has been deemed insecure for a long long time. As you noted yourself, there are a lot of information on how to crack WEP keys online, and it's (almost) as simple as running a terminal command. Anyone can do it. Remember that in most countries, you are liable for what passes through your network. So if an adversary downloads torrents, children pornography or bomb schematics, you will be held liable. Is prison better than spending US$20 on a new router?

There are two possible solutions here, the first might work, the second definitely works.

What might work:

1. Change your password

2. Set up MAC address filtering:

Note that, if your neighbour is tech-savvy enough, he or she might be able to edit or "spoof" his MAC address. This way, the neighbour can "pretend" to be you, by authenticating with the router using your computers MAC address.

What definitely works.

Get a new router. Seriously. WEP was deprecated in 2004 and has been deemed insecure for a long long time. As you noted yourself, there are a lot of information on how to crack WEP keys online, and it's (almost) as simple as running a terminal command.

##Note: Changing your password to a loooong one, doesn't make any difference.

The reason why changing to a longer password won't work, is because of how WEP uses RC4 to encrypt the data.

You can read more about it on Wikipedia.

There is really only one solution to your problem. I do note however that you're not interested in upgrading your router, so I will talk a little about that.

Remember that by not upgrading your router, you are only delaying the inevitable.

What will work long enough for you to get a new router:

- PULL THE PLUG

This is honestly the best solution untill you get a new router.

- or TURN OFF THE WIFI

You can probably do this from the administration panel of your router, if not, remove the antennas from the physical device and go ethernet only. If that is not possible, unplug.

- or CHANGE YOUR WEP PASSWORD and TURN ON MAC FILTERING

Why this won't work in the long run:

MAC filtering is fairly simple for someone to who's a bit tech savvy to avoid. There are multiple tutorials on the subject, and it's really as simple as running a few terminal commands. Since your neigbour has already cracked your WEP-key, chance is, he has the skills enough to google how to spoof a MAC address. 

This way, the neighbour can "pretend" to be you, by authenticating with the router using your computers MAC address.

Changing the password won't last for long, because of how WEP works. An attacker only need to listen to your network for a short period of time, to be able to extract your password. You can read more about it on Wikipedia.

And at last, since you've been breached, anything you have connected to your access point could in theory be infected now. Not only could the hacker have changed your DNS settings, started logging network activity etc, he could also now be in the possession of all your files, pictures, passwords, credit-card information etcetera. To be honest, all of your devices (if not already hacked) are in great danger of getting breached any second.

What you have to do:

Get a new router. Seriously. WEP was deprecated in 2004 and has been deemed insecure for a long long time. As you noted yourself, there are a lot of information on how to crack WEP keys online, and it's (almost) as simple as running a terminal command. Anyone can do it. Remember that in most countries, you are liable for what passes through your network. So if an adversary downloads torrents, children pornography or bomb schematics, you will be held liable. Is prison better than spending 20$ on a new router?

[Edit 07.07.15 13:18] I have rephrased the answer to more correctly state the fact that a new router is the only long term option, as noted by @JamesRyan