For $X$ a pointed topological space, its suspension spectrum $\Sigma^\infty X$ is the spectrum given by the pre-spectrum whose degree-$n$ space is the $n$-fold reduced suspension of $X$:
(e.g. Elmendorf-Kriz-May, example 1.1)
As a symmetric spectrum: (Schwede 12, example I.2.6)
See at Omega spectrum – Completion of a suspension spectrum.
As an infinity-functor $\Sigma^\infty\colon Top_* \to Spec$ the suspension spectrum functor exhibits the stabilization of Top.
The suspension spectrum functor is strong monoidal.
On the one hand, this is the case for its incarnation as a 1-functor with values in structured spectra (this Prop.) Via the corresponding symmetric monoidal model structure on structured spectra this exhibits strong monoidalness also as an (infinity,1)-functor.
More abstractly this follows from general properties of stabilization when regarding stable homotopy theory as the result of inverting smash product with the circle, via Robalo 12, last clause of Prop. 4.1 with last clause of Prop. 4.10 (1). For emphasis see also Hoyois 15, section 6.1, specifically Hoyois 15, Def. 6.1.
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Frank Adams, part III, section 2 of Stable homotopy and generalised homology, 1974
Anthony Elmendorf, Igor Kriz, Peter May, example 1.1 of Modern foundations for stable homotopy theory, in Ioan Mackenzie James, Handbook of Algebraic Topology, Amsterdam: North-Holland (1995) pp. 213–253, (pdf)
Nicholas J. Kuhn, Suspension spectra and homology equivalences, Trans. Amer. Math. Soc. 283, 303–313 (1984) (JSTOR)
John Klein, Moduli of suspension spectra (arXiv:math/0210258, MO)
Stefan Schwede, Example I.2.6 in Symmetric spectra, 2012 (pdf)
Suspension spectra of infinite loop spaces are discussed (in a context of Goodwillie calculus and chromatic homotopy theory) in
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