Fernando Muro's comment is right on the money. I'm interested in this type of question, too, and have read Hovey's paper closely. I'm going to summarize the paper here. I believe Marc Hoyois is correct that you don't need much machinery to make this work in $(\infty,n)$ categories (see Dylan Wilson's comment here), so I'll focus on model categories with an eye towards answering your question about hypotheses. The process of going from $M$ to the stable model structure on $Sp^{\mathbb{N}}(M,G)$ or on $Sp^{\Sigma}(M,G)$ is called stabilization.
Going from the model category of spaces to the model category of spectra, the idea is to make the endofunctor $\Sigma$ into a Quillen equivalence. So Hovey takes a model category $M$ and an endofunctor $G$, then constructs a model category $Sp^{\mathbb{N}}(M,G)$ whose objects are sequences $(X_n)$ along with structure maps $GX_n\to X_{n+1}$. You can endow this category with the projective model structure (i.e. weak equivalences and fibrations are defined levelwise) whenever $M$ is cofibrantly generated. This is the only hypothesis you need on $M$. As for $G$, you need to know it's a left Quillen functor from $M$ to $M$.
In order to make $G$ into a left Quillen equivalence on $Sp^{\mathbb{N}}(M,G)$, you want to do Bousfield localization. So you need this category to be left proper and cellular or left proper and combinatorial (or do you? See my most recent MO question). It turns out that if you assume these properties on $M$ then they hold on $Sp^{\mathbb{N}}(M,G)$. With this assumption, you can create a stable projective model structure on $Sp^{\mathbb{N}}(M,G)$ which makes $G$ into a left Quillen equivalence. Furthermore, if $G$ was already a left Quillen equivalence then the embedding $M\to Sp^{\mathbb{N}}(M,G)$ is a left Quillen equivalence. So $Sp^{\mathbb{N}}(M,G)$ is initial in some sense with respect to the property that $G$ becomes a left Quillen equivalence. You don't have a notion of stable homotopy groups in $Sp^{\mathbb{N}}(M,G)$, but if you want one you can read section 4 of Hovey's paper where he tries to find the correct hypotheses to make this $Sp^{\mathbb{N}}(M,G)$ because more like the usually category of (topological) spectra.
For the monoidal situation you should start with a monoidal model category $M$ with a monoidal left Quillen endofunctor $G$. More generally, let $M$ be a $D$-model category and let $G$ be a left $D$-Quillen endofunctor (you recover the monoidal case for $D=M$). Then Theorem 5.7 shows that $Sp^{\mathbb{N}}(M,G)$ is a $D$-model category and $G$ is a left Quillen equivalence on it provided we know that $G(X\otimes K) = GX\otimes K$ coherently for $X\in M$ and $K\in D$, that $M$ satisfies the properties above, and that $D$ is cofibrantly generated with domains of the generating (trivial) cofibrations being cofibrant. This hypothesis on $D$ appears in a lot of Hovey's work. One small result in my thesis obtains this hypothesis from more standard hypotheses. I can edit or comment with details on that if you're really interested.
Section 6 of Hovey's paper shows you how to do symmetric spectra $Sp^{\Sigma}(M,G)$. Now $G$ is again a left $D$-Quillen endofunctor on $M$, so it must have the form $G(X) = X \otimes G(S)$ where $S$ is the unit of $M$. Now you only need to assume $M$ and $D$ are left proper and cellular, and that $G(S)$ is cofibrant. You again get the projective model structure on $Sp^{\Sigma}(M,G)$ and you can again take Bousfield localization to get the stable version. Theorem 7.11 shows you that everything works out (i.e. $Sp^{\Sigma}(D,G)$ is a monoidal model category and $Sp^{\Sigma}(M,G)$ is a $Sp^{\Sigma}(D,G)$-model category) provided that the domains of the generating (trivial) cofibrations of $M$ and $D$ are cofibrant. The category $Sp^{\Sigma}(M,G)$ is again initial and satisfies other nice properties, as you can see in Sections 8 and 9.
Last comment: I don't know what these hypotheses reduce to in the situation of $R$-modules and $R$-algebras. I'd be interested in thinking about that, especially regarding this hypothesis that makes the domains of the generating maps be cofibrant.
