The vibration-based Structural Health Monitoring plays a central role in ensuring the safe operation of infrastructures by monitoring their structural integrity based on data collected by sensors. While damage detection has reached maturity, the localization and the quantification of small-scale damage remain an open challenge. To address it, both the localization and the quantification of damage are often posed as an updating problem of a Finite Element Model (FEM) of the operating structure, minimizing the misfit between some features computed from response measurements of a faulty structure and its FEM in a reference, healthy condition. This paper investigates the choice of the features for the design of the objective function to quantify structural cracks. For this purpose, a FEM of a beam with a transverse crack is developed and parametrized by the second moment of area of the elements to locate and quantify the crack-related damage. Subsequently, the impact on the choice of the objective function is discussed based on a small-samples Monte Carlo study.