For a given nonlinear operator \( F: X \supset \text{dom } F \to Y \) with \( X, Y \) being Hilbert spaces, the equation \[ F(x) = y, \tag{1} \] is considered which is supposed to be ill-posed in general, and the right-hand side \( y \in Y \) is approximately known only, i.e., \( y_\delta \in Y \) and \( \delta > 0 \) are given such that \( |y_\delta - y |\leq \delta \) holds. The following generalized Tikhonov regularization for equation (1) is considered, \[ |F(x) - y_\delta |^2 + \alpha |Bx - Bx^* |^2 \to \min, \qquad \quad x \in \text{dom } F \cap X_s. \tag{2} \] Here \( B: X \supset \text{dom } B \to Z \) is a closed linear operator, where \( Z \) is a Hilbert space and \( \text{dom } F \subset \text{dom } B \) holds. Moreover, \( (X_s)_{s \in \mathbb{R}}\) denotes a Hilbert scale which is generated by some operator in \( X \), \( \alpha > 0 \) is a small regularization parameter, and \( x^* \in \text{dom } B \) is some initial guess for a solution \( x_0 \in \text{dom } F \) of equation (1). For fixed \(\alpha>0\) and \(\delta > 0\), the existence of a minimizing element for (2) is discussed, and the stability of the minimization problem (2) with respect to perturbations of \( y_\delta \) is investigated. Moreover, under some additional assumptions error estimates are given for approximate minimizers of problem (2) provided that the regularization parameter is chosen appropriately. The obtained results are applied to a parameter estimation problem.