What is the term for a component of a quantity's units? Imagine a company pays for a service for each employee.  The service costs $10/employee/month.  Written another way, the cost is "10 dollars per employee per month."

My question focuses on 10 dollars per employee per month and finding a name for some of the parts or pieces of this phrase.  I've been calling 10 the value and dollars per employee per month the unit (or units).  If we further examine the units, dollar seems to hold different meaning than employee and month.

What term can I use to describe dollar and what term can I use to describe employee or month?

So far I've come up with "unit rate" (but that ignores that there may be an additional rate "month") and "dimension".
This is more of a scientific question than linguistic, but "dollars per employee per month" is a perfectly good unit. It is not fundamentally different from "miles per hour" or "metres per second squared" (a common unit of acceleration). Another similar usage of unit might be "dollars per person" or "dollars per capita", which are commonly used to measure relative GDP, for example.

"Dimension" has a different meaning in the calculation of units and refers to different types of units - for example Length, Time or Money (Dimension 'Length' for example has units such as metre, inch or furlong)

If you want to differentiate between units 'above' and those 'below' the line, try the terminology of fractions - i.e. 'numerator' and 'denominator'.
Dollars, employees and months are all units.  Dollars per employee per month is also a unit.  You can distinguish between these by calling "dollars" a base unit or a fundamental unit.  Dollars per employee per month is a composite unit or a derived unit.

Turning dollars per employee per month into, for instance, dollars per team per workday is performing a base unit conversion, just the same as converting miles per hour into meters per second is a base unit conversion.  The metric remains the same (money over staffing by time, or distance over time) but the value changes as the units change.

There is a difference between base and fundamental.  The SI standard establishes one fundamental unit for each base.  Any unit of time, for instance, is a base unit -- centuries, years, months, days, whatever.  Time cannot be expressed as a composite of other units.  The fundamental unit of time in SI is the second.  If you're not certain which one unit of a base metric counts as the fundamental unit, just use the phrases "base unit" and "composite unit".  

It's true that base units measure single dimensions.  Unfortunately, the word "dimension" can be confusing.  Time counts as a dimension.  So do money and staffing (although those aren't dimensions in physics).  It's far simpler to speak of units and metrics, and leave the confusion of dimensions unmentioned.