This phrase suggests that someone has the power or authority to grant something within their capacity. The allocation of funds is in the gift of the finance director.
Pro Tip 2/3
Exclusive Right
Implies that only the person or entity mentioned has the exclusive right to give or withhold the particular thing. Admission to the elite club is in the gift of its members.
Pro Tip 3/3
Not Transferable
The control to grant something is not usually transferable; it is specific to the person or role. The scholarships are in the gift of the department chair alone.