You would not believe how much consternation this simple question has caused my office. :)
When installing cage nuts to mount rails or equipment, the nuts are clipped into a piece of metal with square holes, usually labeled with the rack unit number:
... as highlighted in red.
How does one refer to this part of a rack?
Mounting Flange. It's defined in "EIA-310-D, Cabinets, Racks, Panels, and Associated Equipment". It may not be easy to find a legitimately free (the EIA charges for it) copy online, but you can find plenty of references to it.
I've never heard a specific name for only that exact part of the rack.
Like @joeqwerty said, usually it's just called the post. A standard server rack is called a "4-post rack", and there are "2-post racks" which were (are?) common in telco use when the gear to be mounted is a bit lighter in weight.
However, that really refers to the whole piece of metal, not just the part which the cage nuts attach to. So @luke-peterson's suggestion of "cage" is probably the closest thing to just the part you're trying to name.