You cannot add internet functionality to a DS Lite game that lacks built-in Wi-Fi support. The hardware and software limitations make this impossible.
For Wi-Fi compatible games, the Wiimmfi project provides custom DNS servers that restore online multiplayer, leaderboards, and DLC. Requires changing DNS settings in the DS connection setup—no hardware mods needed.
Bottom line: Online capability is determined per game at development time. Choose Wi-Fi enabled titles if online play is a priority.