I click "Restrict renderability", so that everything except the item I want to export is grayed out. However, it still exports lights and everything when I import it into Unity.
How can I fix this?
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The render visibility option only affects what is visible when blender renders a final image, it isn't used when exporting objects.
If you give unity your blend file, it can do the export/import for you and shouldn't import lights and cameras, see the unity manual for the full list.
If you are exporting manually, you should find options to control what is exported, the fbx exporter has an option that lets you turn off the export of cameras and lights, while the obj exporter can be set to export only selected objects.