This paper investigates the finiteness properties of generalized local cohomology modules. Let \(R\) be a commutative Noetherian ring with identity, and \(\mathfrak{a}\) an ideal of \(R\). For two \(R\)-modules \(X\) and \(Y,\) and an integer \(i\geq 0\), the \(i\)th \textit{generalized local cohomology} module of \(X\) and \(Y\) with respect to \(\mathfrak{a}\) is defined by \[ \text{H}_{\mathfrak{a}}^i(X,Y):=\underset{n\in \mathbb{N}}{\varinjlim}\ \text{Ext}_R^i(X/\mathfrak{a}^nX,Y). \] Note that the usual local cohomology modules \(\text{H}_{\mathfrak{a}}^i(Y)\) correspond to the case where \(X=R\). Since Robin Hartshorne introduced the concept of cofiniteness, it has become a central topic in the study of finiteness properties of local cohomology modules. An \(R\)-module \(X\) is said to be \(\mathfrak{a}\)-\textit{cofinite} if \(\text{Supp}_R(X) \subseteq \text{V}(\mathfrak{a})\), and the \(R\)-module \(\text{Ext}_R^i(R/\mathfrak{a},X)\) is finitely generated for all \(i\geq 0\). There are two generalizations of this notion by using minimaxness and weakly Laskerianness. An \(R\)-module \(M\) is called \textit{minimax} if it contains a finitely generated submodule \(N\) such that the quotient module \(M/N\) is Artinian. It is called \textit{weakly Laskerian} if every quotient of \(M\) has only a finite number of associated primes. An \(R\)-module \(X\) is called \(\mathfrak{a}\)-\textit{cominimax} (respectively, \(\mathfrak{a}\)-\textit{weakly cofinite}) if \(\text{Supp}_R(X)\subseteq \text{V}(\mathfrak{a})\), and the \(R\)-module \(\text{Ext}_R^i(R/\mathfrak{a},X)\) is minimax (respectively, weakly Laskerian) for all \(i\geq 0\). It is straightforward to see that every minimax module is weakly Laskerian, and hence every \(\mathfrak{a}\)-cominimax module is also \(\mathfrak{a}\)-weakly cofinite. This paper extends some known results on \(\mathfrak{a}\)-cofiniteness of local cohomology modules to the notions of \(\mathfrak{a}\)-cominimaxness and \(\mathfrak{a}\)-weak cofiniteness in the setting of generalized local cohomology modules. The main results are as follows: \vspace{0.2cm} \textbf{Theorem A:} Let \(X\) be a finitely generated \(R\)-module, and let \(Y\) be an \(R\)-module such that \(\text{H}_{\mathfrak{a}}^i(Y)\) is \(\mathfrak{a}\)-cominimax for all \(i\geq 0\). Then the \(R\)-module \(\text{H}_{\mathfrak{a}}^t(X,Y)\) is an \(\mathfrak{a}\)-cominimax for all \(t\geq 0\), provided one of the following conditions holds: \begin{itemize} \item[(a)] \(\dim_R(X)\leq 2\); or \item[(b)] \(\dim_R(\text{H}_{\mathfrak{a}}^i(Y))\leq 1\) for all \(i\geq 0\). \end{itemize} \textbf{Theorem B:} Assume that \(R\) is local. Let \(X\) be a minimax \(R\)-module, and let \(Y\) be an \(R\)-module such that \(\text{H}_{\mathfrak{a}}^i(Y)\) is \(\mathfrak{a}\)-cominimax for all \(i\geq 0\). Then the \(R\)-module \(\text{H}_{\mathfrak{a}}^t(X,Y)\) is \(\mathfrak{a}\)-weakly cofinite for all \(t\geq 0\), provided one of the following conditions holds: \begin{itemize} \item[(a)] \(\dim_R(X)\leq 3\); or \item[(b)] \(\dim_R(\text{H}_{\mathfrak{a}}^i(Y))\leq 2\) for all \(i\geq 0\). \end{itemize}