This could have been a comment to Peter's answer, except that it's going to be long. (Maybe I should be making it a new question.)

Suppose that $Ho(\mathcal C)$ is obtained from $\mathcal C$ by (universally) inverting a class $\mathcal W$ of morphisms. Let's call an object of $\mathcal C$ homotopically terminal if in $Ho(\mathcal C)$ it is terminal. 

I did not assume anything about $\mathcal W$ (except that this localization existed). In practice we usually have more to work with, like a model structure or at least the 2 out of 6 condition. Given enough structure or information, there are ways of defining a simplicial set for each pair of objects such that this serves as a "function space", in particular such that its set of components is the set of morphisms in $Ho(\mathcal C)$. I'm not a master of the known sufficient conditions for making these function spaces -- I suppose that there are several overlapping approaches. 

QUESTION: Can we say that (in any or all of these approaches) an object $X$ is homotopically terminal in the sense I gave above if and only if for every object $Y$ the function space $Hom(Y,X)$ is weakly equivalent to a point? I hope so.

Here is a general, relatively simple, approach to the idea of derived functor, really just a distillation of standard ideas as I understand them, but trying to steer away from extraneous optional technicalities:

Suppose that in $\mathcal C$ and in $\mathcal D$ there are classes of morphisms $\mathcal U$ and $\mathcal V$ respectively. Let $\mathcal F$ be the category of functors $\mathcal C\to \mathcal D$. In $\mathcal F$ let $h\mathcal F$ be the full subcategory of "homotopy-invariant" functors -- those which take $\mathcal U$ into $\mathcal V$. By a left derived functor of $F\in \mathcal F$ I mean the following: Form the category $hF$ of all homotopy-invariant functors over $F$. An object is a pair $(G,G\to F)$ where $G\in h\mathcal F$. A morphism is a map $G_1\to G_2$ compatible with the maps to $F$. Define the class of maps $\mathcal W$ in $hF$ by requiring that for every $X\in \mathcal C$ the map $G_1(X)\to G_2(X)$ belongs to $\mathcal V$. By a left derived funtor of $F$ let us mean a homotopically terminal object of the category of $hF$ (with respect to this class $\mathcal W$).

The standard way to achieve such a thing is to use a "replacement functor" $Q:\mathcal C\to \mathcal C$. You need a map $Q\to 1$, and you need three things to be true: (1) $QX\to X$ always belongs to $\mathcal U$. (2) $Q$ takes $\mathcal U$ into $\mathcal U$. (3) As applied to maps between objects that are in the image of $Q$, $F$ takes $\mathcal U$ into $\mathcal V$. I believe that that's all you need to show that $F\circ Q$ is homotopically terminal in the sense I defined. The proof is basically contained in what Peter wrote, including the edits prompted by my comments.

You don't even need the 2 out of 3 condition for $\mathcal U$ or $\mathcal V$ to get this. 

Right?