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.