When it comes to a manager presenting a case for expanding the employee ranks, they need to be able to speak to the company's bottom line. The most important question to ask, answer, and present in the affirmative is: "Is the company losing money by not having this hire?"
Quantifying the Need
Part of the answer to this question involves calculating the cost of having overworked current employees. For example, if the environment is one of a help desk, it can be very easy to show how, on an hourly basis, the absence of one or more additional team members is causing the support staff to spend costly, hourly waged time dealing with issues that a new, junior employee could take off their watch, thereby freeing them up for more senior, experienced matters.
Another way of asking the same question is: "Is the company missing out on potential revenues by not having this hire?" Answering this one involves a bit more forward thinking, as well as, usually, an evaluation of what direct competitors in the space are up to. However, if a manager can show that a major new direction being contemplated by a competitor will fail to be properly handled without adequate new staffing, they can not only fill a need but also prove they are a pro-active, big picture team member themselves.
In fact, that big picture is one of the best ways to present the need for a new hire. On a granular level, any new hire is seen by a supervisors as an added, baseline cost. However, skilled managers will present the need within the context of the company's tactical and strategic needs.
It is, on the one hand, about bringing in more additional revenue than new-hire costs. But it is also about supporting the company's long-range strategic goals. The cost of a mission critical failure is, in many cases, larger than the simple dollar cost. So tying the new hire to a department's ability to properly avoid that kind of pitfall is another great way to justify the need.