One way is to turn the visibility off for all the other layers. go to Image>Trim>Transparent Pixels and then save for web.
Then you can undo the trim step in the larger file if you want to continue working on it.
Thanks R_Kelly - that was easy!
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Or Ctrl/Cmd click that layer,
Ctrl/Cmd c to copy it,
Ctrl/Cmd n to create a new document of the size of the image now in the clipboard
Ctrl/Cmd v to paste the image from clipboard to the new image.
Thanks Trevor - that works well also!
If you do a lot of saving graphic elements from web page mock-ups, it's worth reading up on using slices.
Also have a look at layer comps. Combining the two enables you to save all the graphic elements of the web page (including various roll-over states) from one master file.