@Chris L. I think I need to object to the idea on philosophical grounds.
When you extract a late fee it's because the tenant has caused you legal injury because if their failure to pay on time. If you "discount the rent" for paying early, respectfully, it feels like you've transitioned from landlady to land-nanny.
Vacancy is a bigger threat to your bottom line than mild tardiness with respect to payment. I know some landlords that take a lesson from cell phone shops: they use inexpensive loss leaders to secure longer term contracts.
How this usually works:
Someone signs up for a lease with a term of 14 (or 18) months instead of the more typical 12 months. In exchange they're given a loss leader such as a 50" TV or an iPad for "free." The value is agreed upon contractually in a lease addendum and is usually more than the landlord paid e.g., a $250 TV with a contractually agreed upon "value" of $600. If the tenant breaks the lease the value in the addendum is added to any future collection attempt.
If done correctly, in the right area, this can decrease the amount of turnover and increase the speed with which you take on new tenants in the event of a vacancy. What do you think?