Here are a few points to bear in mind:
Never "threaten to quit" unless you actually intend to quit: Your friend should only threaten to quit if she genuinely intends to quit if her requirements are not met. She should have a definite decision in mind in advance, including a timeline for how long she is willing to wait. If this is a course of action she is considering, she should already be looking for alternative positions elsewhere and ideally she would have other oportunities lined up.
Don't do charity for an institution that doesn't value you: You mentioned that your friend is reluctant to quit because "...if she quits the group would be in good amount of trouble (no other person to guide the PhDs, etc.)". That is tough luck for the institution. If they aren't going to pay her what she is worth then they can take the loss that comes with losing her. It is good for an employee to be aligned with the goals of the institution, and want to act in its best interests, but that should be a two-way street.
Be cognisant of the "principal-agent" problem: The principal-agent problem is a well-known problem in economics and institutional analysis. Your university is the principal and the HR representative is its agent. So long as the HR representative gets to exercise her power over her little fiefdom, she might not give a shit if her decision leads to loss of a good staff member to the university. It is unlikely that she will pay any price for this if it occurs and it is unlikely that the university will penalise her, so she may act in her own interests and not necessarily the interests of the university. In view of that, you should bear in mind that any threat to quit, while possibly damaging for the principal, probably has no adverse impact on the agent, so it is likely to be ignored.