What interesting homotopy invariants can I write down using the universal property of homotopy types? I've recently been led to believe some version of the following statement: 

Weak homotopy types, or equivalently $\infty$-groupoids (let me not commit myself to a particular model of these), are freely generated under homotopy colimits by a point in the sense that, if $C$ is an $(\infty, 1)$-category with small homotopy colimits, then the $(\infty, 1)$-category of homotopy colimit-preserving $(\infty, 1)$-functors $\infty\text{-Gpd} \to C$ should be equivalent to $C$, with the equivalence on objects being given by evaluating the functor at a point. 

This is the $(\infty, 1)$-categorical version of the $1$-categorical statement that $\text{Set}$ is freely generated under colimits by a point. 
So we should be able to construct weak homotopy invariants by finding nice $(\infty, 1)$-categories and nice objects in them and seeing what the corresponding functor above produces. I think the following are examples, although the functors will be contravariant and I could have the details wrong:


*

*Take $C$ to be the ($(\infty, 1)$-category presented by) the dg-category of complexes of abelian groups, and take the object assigned to the point to be $\mathbb{Z}$. The corresponding weak homotopy invariant should be singular cochains on a space. This should be some version of Eilenberg-Steenrod. 

*Take $C$ to be the $(\infty, 1)$-category of dg-categories, and take the object assigned to the point to be the dg-category of complexes of abelian groups. The corresponding weak homotopy invariant should be the dg-category of $\infty$-local systems (with values in complexes of abelian groups) on a space.



What other interesting weak homotopy invariants can be described this way?

 A: Here is a slight generalization of what you are thinking about, related to Thom spectra (disclaimer, I am still learning this stuff, so, someone correct me if I make a mistake):
Let $R$ be some $A_\infty$-ring spectrum. Let $R\mbox{-line}$ denote the connected space (infinity group) of self (homotopy) automorphisms of $R$ in the $(\infty,1)$-category of $R$-modules. Note that there is a canonical functor $$\theta:R\mbox{-line} \to R\mbox{-Mod}.$$ Also note that there is a canonical equivalence of $(\infty,1)$-categories $$\mbox{Fun}(R\mbox{-line}^{op},\infty Gpd) \simeq \infty Gpd/R\mbox{-line}.$$ An object on the right may be thought of as a generalized spherical fibration, since if $R$ is the sphere spectrum, (stable) spherical fibrations are classified by maps into $BGl_1 R = R\mbox{-line}.$
The functor $\theta$ may be extended to a unique (homotopy) colimit preserving functor $${Th}_R:\mbox{Fun}(R\mbox{-line}^{op},\infty Gpd) \simeq \infty Gpd/R\mbox{-line} \to R\mbox{-Mod},$$ such that ${Th}_R$ restricted to the Yoneda embedded image of $R\mbox{-line}$ is just $\theta$. A simple calculation shows that in fact ${Th}_R$ sends a map (functor) $$f:X \to R\mbox{-line}$$ to the (homotopy) colimit of $\theta \circ f.$ This colimit is the generalized Thom spectrum of $f$. E.g., if $R$ is the sphere spectrum, and $M$ is a manifold with its canonical spherical fibration $$T_M:M \to R,$$ coming from the image of its tangent bundle under the $J$-homomorphism, then $Th_R\left(T_M\right)$ is the Thom spetrum of $M$. More generally, for the $J$-homomorphism itself $$J:BU \to BGL_1S,$$ $Th\left(J\right)$ is $MU.$
