You can simply get a debit card from your bank and not have to run your credit.

Yes, I've never had this situation, and it seems as though the would-be renter somehow thinks I find the idea of prepaid rent appealing enough for me to drop the rent price.

Majority of real estate syndications will not be able to service their debit and be in serious trouble very soon.   

If you are NOT tracking tenants as customer then simply do a GJE ("general journal entry" and enter the received payment, lets assume its a $500 check, as such:Account- Saving Account   ;  Debit: $500 ;            Memo- Security Deposit Apt#1 Mr.

Maybe the tenant is trying to prove that she can pay a years' rent (yes, I'm being optimistic here, but not ignoring the potential pitfalls...)Say that you don't accept pre-paid rent for accounting and tax reasons, but you would be happy to see a "proof of funds" in the way of 2-3 months bank statements.

Also consider that this tenant has a tad bit lower credit score than I'd prefer but pre paid a couple of months worth of rent already. 

In that case, I will use the prepaid rent to cover the monthly rent loss but once I have the property rented and say I have a balance of 3 months of rent left, does the remaining prepaid rent money go back to the tenant, if I find them?

I then let the surplus bank as extra pre-paid rent as they get back on their feet.The small raise also becomes an expected annual thing.  

This includes turning over pre-paid rent and security deposits to the new owner.

-prepayment charge on a prepaid amount over 20% of the original mortgage in any 12 month period, can only be a max of 6 months advance interest on that amount (over 20% of original mortgage).https://leginfo.legislature.ca.gov/faces/codes_displaySection.xhtml?