Open Settings → Profile. Under Profile picture, choose one of your own uploaded Flipnotes and save the page.
The website uses that Flipnote's selected thumbnail frame. The DSi Creator's Room avatar is made from the same desktop PNG source, center-cropped and resized for the DSi profile slot, so the two versions use the same chosen artwork.
Automatic (latest Flipnote) leaves the explicit avatar setting empty. In that mode, RexiMemo uses your newest Flipnote as the current profile picture when one is available.