What factors should be considered when taking private pilot renter's insurance?
Ignoring what you should and should not have in terms of coverage for your self there are some facts to note about getting gap insurance. 

You need to speak with your flight school and ask them what they require you to have as every flight school could be insured differently. For example, when I got my renters insurance I had to have 10K of aircraft damage coverage as the flight schools insurance did not kick in until then. Had anything happened I would have been responsible for the 0-10K worth of damage. In this case you would want coverage for damage to the aircraft. If the flight school has insurance on any and all damage you may not need it on your insurance.