It is related to the philosophical notion of the same name in that these predictions describe anticipated behaviors for the phenomena in reality.
Phenomenology is commonly applied to the field of particle physics, where it forms a bridge between the mathematical models of theoretical physics (such as quantum field theories and theories of the structure of space-time) and the results of the high-energy particle experiments.
Within the well-tested and generally accepted Standard Model, phenomenology is the calculating of detailed predictions for experiments, usually at high precision (e.g., including radiative corrections).
Phenomenological analyses, in which one studies the experimental consequences of adding the most general set of beyond-the-Standard-Model effects in a given sector of the Standard Model, usually parameterized in terms of anomalous couplings and higher-dimensional operators.
In this case, the term "phenomenological" is being used more in its philosophy of science sense.