What's a good synonym for "unwantedly"?

Most people would probably say accidentally, but personally I'd be likely to use inadvertently in most similar contexts.

In this exact context, the intention is to place the "blame" for the error on the design of the webapp, rather than the user's carelessness. Personally I feel that inadvertently doesn't do that quite so well as erroneously, by mistake, or mistakenly, all of which seem to me more capable of implying that the user was "led astray" rather than inherently incompetent.

If "Cancel" button is misleadingly labelled/displayed, and the user instantly realises he's selected the wrong thing as soon he clicks, it might be reasonable to say unwittingly.

You can use the word inadvertently.