Nintendo has said that it's "looking into" allowing people to transfer purchases from one DSi to another.
"We're looking into that specific topic, but we don't have anything to announce at this time," a spokesperson told Kotaku.
Currently, content bought from the DSi Shop cannot be re-downloaded on another handheld, nor transferred between consoles.
While some consoles and devices built around digital distribution restrict the number of transfers, or force you to jump through hoops, most at least allow you to re-download content.
When they don't - as with SingStar purchases - it's usually possible to claw your money back from the content provider in the form of a store credit.