The stabilization of an (∞,1)-category $C$ with finite (∞,1)-limits is the free stable (∞,1)-category $Stab(C)$ on $C$. This is also called the $(\infty,1)$-category of spectrum objects of $C$, because for the archetypical example where $C =$ Top the stabilization is $Stab(Top) \simeq Spec$ the category of spectra.
There is a canonical forgetful (∞,1)-functor $\Omega^\infty : Stab(C) \to C$ that remembers of a spectrum object the underlying object of $C$ in degree 0. Under mild conditions, notably when $C$ is a presentable (∞,1)-category, this functor has a left adjoint $\Sigma^\infty : C \to Stab(C)$ that freely stabilizes any given object of $C$.
Going back and forth this way, i.e. applying the corresponding (∞,1)-monad $\Omega^\infty \circ \Sigma^\infty$ yields the assignment
that may be thought of as the stabilization of an object $X$. Indeed, as the notation suggests, $\Omega^\infty \Sigma^\infty X$ may be thought of as the result as $n$ goes to infinity of the operation that forms from $X$ first the $n$-fold suspension object $\Sigma^n X$ and then from that the $n$-fold loop space object.
Let $C$ be an (∞,1)-category with finite (∞,1)-limit and write $C_* := C^{{*}/}$ for its (∞,1)-category of pointed objects, the undercategory of $C$ under the terminal object.
On $C_*$ there is the loop space object (infinity,1)-functor $\Omega : C_* \to C_*$, that sends each object $X$ to the pullback of the point inclusion ${*} \to X$ along itself. Recall that if a $(\infty,1)$-category is stable, the loop space object functor is an equivalence.
The stabilization $Stab(C)$ of $C$ is the (∞,1)-limit (in the (∞,1)-category of (∞,1)-categories) of the tower of applications of the loop space functor
This is (StabCat, proposition 8.14).
The canonical functor from $Stab(C)$ to $C_*$ and then further, via the functor that forgets the basepoint, to $C$ is therefore denoted
Concretely, for any $C$ with finite limits, $Stab(C)$ may be constructed as the category of spectrum objects of $C_*$:
This is definition 8.1, 8.2 in StabCat
Given a presentation of an (∞,1)-category by a model category, there is a notion of stabilization of this model category to a stable model category. That this in turn presents the abstractly defined stabilization of the corresponding (∞,1)-category is due to (Robalo 12, prop. 4.14).
If $C$ is an $(\infty,1)$-category with finite limits that is a presentable (∞,1)-category, then the functor $\Omega^\infty : Stab(C) \to C$ has a left adjoint
Prop 15.4 (2) of StabCat.
stabilization is not in general functorial. It’s failure of being functorial, and approximations to it, are studied in Goodwillie calculus.
A general discussion in the context of (∞,1)-category theory is in section 1.4 of
Discussion of the relation between stabilization of (∞,1)-categories (to stable (∞,1)-categories) and of model categories (to stable model categories) is in section 4.2 of