Galaxian
Lurker
Like I said, the Lite could only log in on secure networks that used a WEP encryption, and most routers nowadays come with WPA as a default. You would have had to go into the router's setting and manually change it to WEP.
DSi, DSi XL, 3DS, and 3DS XL are all compatible with both types of encryption, so you shouldn't have a problem buying a 3DS and getting on the network.