I want to power some accessories in my car using illuminated switches on the dashboard. These switches will be wired to the dash illumination circuit, which is normally 12V but can drop as low as 6V when fully dimmed. Can I use a 6V relay here to give 12V power to the accessories? According to this data sheet, it's max switching voltage is 16V so I think this should be safe.
The coil and contacts of a relay are electrically isolated from each other. Powering the coil with mechanically actuate the contacts. The voltages can differ between the two (e.g. 24VDC coil with 230VAC contacts).
The coil requires a specific voltage to turn the relay on and off. This voltage is selected on page 4 of the document you've provided. Looks like your options are 6, 9, 10, and 12 VDC. EDIT: It would be in the best interest that this coil voltage is not fluctuating, and instead, a constant voltage source. Powering a 6 VDC coil from a 12 VDC source will cause the coil to become quite hot.
The contacts are what will be providing power to your accessories. They have a max rating of 16VDC, 35A (page 3). So, 6 to 12 VDC on your contacts will not be a problem, so long as you have a constant voltage on your coil.