Every time I logged into my Lion machine, Aperture started up.
I checked the Login items option on my user account. Nothing there.
Then I figured it was a bug with OS X 10.7 Lion resume. I deleted all the saved states, including Aperture's (Delete Specific Application Saved States from Mac OS X 10.7 Lion Resume).
Finally, something clued me. This wasn't a new Lion problem, it was the old 'launch Aperture when iPhone connected' bug. Same thing can happen with iPhoto or Image Capture or Preview or "Auto Importer". This particular machine is connected to a USB hub that had some iPhones attached.
I don't know the proper place to control this peculiar OS X behavior, but I do know it can be controlled through Image Capture. I opened that app, and clicked on the iPhones icons on the left side. For each one I set 'Connecting this iPhone' to 'No application'.