The Nintendo DSi doesn't have a traditional "browsing history" like a web browser. Instead, data is stored in discrete categories—DSi Shop downloads, game save data, photos, and SD card contents. Here's how to delete each type, plus the nuclear options.
DSi Menu → "SD Card" → "Format SD Card." Warning: This erases all data on the SD card—photos, music, DSiWare backups, everything. Back up to a PC first if needed.
DSi Menu → "System Settings" (wrench icon) → Page 4 → "Format System Memory." Warning: This deletes everything: DSiWare, saves, photos, settings, Nintendo Wi-Fi Connection config, and Shop account link. The console returns to out-of-box state.
Always back up important data first. Once deleted, DSi data cannot be recovered—there's no cloud sync or recycle bin.