I do understand your difficulty, i.e. in the term "passing rent", just what is it that is actually "passing"? All the definitions available simply provide an explanation of what the term refers to, without actually explaining why it is "passing", and not, say, simply "current" or "existing" or "acutally payable", etc.

It has been extremly difficult to find any explanation at all of the origin of the term, but this is my best shot.

First of all, some more definitions:

passing rent (i.e. the rent that the tenant is currently contracted to pay to the landlord).
http://books.google.co.uk/books?id=fc8CkwyoApAC&pg=PA60&lpg=...

Passing (or contract): The rent specified by a given lease agreement; although a given contract rent may equate to the Market Rent, in practice they may differ substantially, particularly for older leases with fixed rental terms. The term, contract rent is North American usage; passing rent is Commonwealth usage. (IVSC)
http://www.commerce.wa.gov.au/consumerprotection/PDF/Reports...

Passing rent is simply the current contractual rent on the property, expressed in annual terms.
http://www.7dials.com/pages.asp?id=82

The most common form of rent review provision is still what is referred to as an upwards-only review to open market rent. This means that on each designated review date, the rent will be reviewed by comparing the rent actually payable under the lease on that date (the passing rent) with the rent that would be obtained if the Property had been marketed and was being re-let on similar terms on that date (the hypothetical letting).
http://www.practicalconveyancing.co.uk/content/blogcategory/...

I'm not at all sure about this, but believe that "passing" might refer to the fact that the rent or payment "passes" from the tenant to the landlord.

Among the many things that are negotiated, there are several issues that are critical in determining the timing of payments and the amount of cash that passes from the tenant to the landlord during the lease term.
http://books.google.co.uk/books?id=PcwFrMTdFTYC&pg=PA174&lpg...

The tax is based upon the open market value of the property and not the current rent passing between the tenant and the landlord.
http://www.fig.net/commission7/verona_am_2008/papers/12_sept...

I don't think "current rent" on its own is sufficient. Try Googling ""current passing rent" and see how many hits you get.