You might do better using a better antenna on your wireless device. In my never humble opinion, repeaters, range extenders, and range expanders, are evil. Details on request.
Is this a WRE54G v1 or v2? There are some differences. V1 with 1.04.16 will only do WEP which is rather insecure. V2 with 2.01.10 will do both WEP and WPA-PSK (TKIP or AES) which is much better security.
That can only happen if you have a non-unique SSID. Is your neighbor using the default SSID of "Linksys"? If so, kindly convince your neighbor to change it to something unique. My guess(tm) is that the change in networks is caused by another WLAN with the same SSID.
Another way to do it wrong is to use an SSID of "ANY". On some repeaters, this will repeat any SSID that it hears.
There are three items that will prevent unauthorized use in 1.04.16 firmware:
- The SSID.
- The WEP Key.
- The MAC address of the access point. My guess(tm) is that your neighbor has setup their wireless router with all defaults and you have compounded the error by not adding the MAC address.
Actually, it will if your neighbor is running encryption. Reading between the lines, I deduce that he's not running encryption. I'll spare you my standard rant about running a wide open, unsecured, publicly accessible, wireless access point.
With no encryption, that actually might have been a possiblity.