Here is how I look at the situation after thinking hard about the answers. I will say things in the language of triangulated categories and homotopy limits being somewhat imprecise. I suppose some hares would say that this is really about $(\infty,1)$-categories; unfortunately I don't know enough about these gadgets.

We are in the following situation: we have triangulated categories $D_1$ and $D_2$ together with a triangle functor $\alpha: D_1 \to D_2$ (additive, preserving distinguished triangles, etc). We would like to "explain" or "describe" the effect of $\alpha$ on objets. Clearly this necessitates first a way of describing objects in the first place.

The most conventional way I can think of is to use homological functors $\pi_i: D_i \to A_i$ into abelian categories (i.e. functors turning distinguished triangles into long exact sequences). I will write $\pi_{i*}X$ for the graded object $\pi_i(\Sigma^*X),$ where $\Sigma$ is the shift functor. Ideally, we would like a way to compute $\pi_{2*} \alpha X$ in terms of $\pi_{1*}X$ and perhaps some extra data. It is sort of clear that this cannot work in full generality, because $\pi_1$ can throw away arbitrary amounts of information.

So we need a more "internal" way of describing objects. One way to do this is using filtrations. That is, we associate to $X$ a sequence $\dots \to X_{k-1} \to X_k \to X_{k+1} \to \dots$ such that $X = \operatorname{holim} X_\bullet$ or $X = \operatorname{hocolim} X_\bullet.$ For this to be useful we need the "subquotients" $\operatorname{cone}(X_k \to X_{k+1})$ to be "nice", e.g. live in a subcategory we understand well.

The "internal" and "external" approach are not unrelated; for example a t-structure on $D_1$ has as heart an abelian subcategory, every object acquires a filtration and a "cofiltration" (I think), and the "subquotients" live in the abelian subcategory, effectively giving us a compatible $\pi_1.$

So the strategy to describe $\alpha(X)$ is now this:

1. Find the (co)filtration $X_\bullet$ and relate $\alpha(\operatorname{ho(co)lim}X_\bullet)$ to $\operatorname{ho(co)lim}\alpha(X_\bullet).$

2. Relate the subquotients of $X_\bullet$ to the subquotients of $\alpha X_\bullet.$

3. Relate the subquotients of $\alpha X_\bullet$ to $\pi_2\operatorname{ho(co)lim} \alpha(X_\bullet).$

Step one works if $\alpha$ commutes with filtered homotopy limits or colimits (then take the filtration or cofiltration as appropriate). I don't know good conditions for this, but it seems to be common. Certainly (?) homotopy (co)limits commute with homotopy (co)limits, so we can get the spectral sequence of a homotopy (co)limit in this language. Also mapping spaces commute appropriately, so we can get the AH-SS.

Step two really is the input to this entire game. Note that $\alpha$ commutes with finite homotopy (co)limits (i.e. cones) so this is reasonable.

Step three is the spectral sequence of a filtered homotopy type. There is always a spectral sequence, but it may or may not converge to the groups of the homotopy (co)limit (depending on exactness properties of homotopy (co)limits on $A_2$).

I think many spectral sequences can be looked at in this way (certainly Grothendieck, AH, homotopy limit/colimit; probably also Adams).

To expand on Ben Wieland's example of the AH-SS, in this case $D_1 = D_2 = SH,$ $\pi_1 = \pi_2 = \pi$ is the stable homotopy groups, corresponding to the natural t structure, with heart abelian groups, $X$ is a fixed spectrum, $\alpha(E) = Map(X, E).$ Then $\alpha$ sends homotopy limits to homotopy colimits, so we should use the filtration corresponding to $\pi,$ writing $E$ as the homotopy limit of its Postnikov filtration. The subquotients are Eilenberg-Maclane spectra corresponding to the homotopy groups of $E,$ and so we get a spectral sequence

$$ E_2^{*,*} = \pi_*\alpha(\text{subquotients of }E_*) = \pi_* Map(X, H\pi_*E) = H^*(X, \pi_*E) \Rightarrow \pi_*\alpha(E) = H^*(X, E)$$

which is what we wanted.

Here is how I look at the situation after thinking hard about the answers. I will say things in the language of triangulated categories and homotopy limits being somewhat imprecise. I suppose some hares would say that this is really about $(\infty,1)$-categories; unfortunately I don't know enough about these gadgets.

We are in the following situation: we have triangulated categories $D_1$ and $D_2$ together with a triangle functor $\alpha: D_1 \to D_2$ (additive, preserving distinguished triangles, etc). We would like to "explain" or "describe" the effect of $\alpha$ on objets. Clearly this necessitates first a way of describing objects in the first place.

The most conventional way I can think of is to use homological functors $\pi_i: D_i \to A_i$ into abelian categories (i.e. functors turning distinguished triangles into long exact sequences). I will write $\pi_{i*}X$ for the graded object $\pi_i(\Sigma^*X),$ where $\Sigma$ is the shift functor. Ideally, we would like a way to compute $\pi_{2*} \alpha X$ in terms of $\pi_{1*}X$ and perhaps some extra data. It is sort of clear that this cannot work in full generality, because $\pi_1$ can throw away arbitrary amounts of information.

So we need a more "internal" way of describing objects. One way to do this is using filtrations. That is, we associate to $X$ a sequence $\dots \to X_{k-1} \to X_k \to X_{k+1} \to \dots$ such that $X = \operatorname{holim} X_\bullet$ or $X = \operatorname{hocolim} X_\bullet.$ For this to be useful we need the "subquotients" $\operatorname{cone}(X_k \to X_{k+1})$ to be "nice", e.g. live in a subcategory we understand well.

The "internal" and "external" approach are not unrelated; for example a t-structure on $D_1$ has as heart an abelian subcategory, every object acquires a filtration and a "cofiltration" (I think), and the "subquotients" live in the abelian subcategory, effectively giving us a compatible $\pi_1.$

So the strategy to describe $\alpha(X)$ is now this:

1. Find the (co)filtration $X_\bullet$ and relate $\alpha(\operatorname{ho(co)lim}X_\bullet)$ to $\operatorname{ho(co)lim}\alpha(X_\bullet).$

2. Relate the subquotients of $X_\bullet$ to the subquotients of $\alpha X_\bullet.$

3. Relate the subquotients of $\alpha X_\bullet$ to $\pi_2\operatorname{ho(co)lim} \alpha(X_\bullet).$

Step one works if $\alpha$ commutes with filtered homotopy limits or colimits (then take the filtration or cofiltration as appropriate). I don't know good conditions for this, but it seems to be common. Certainly (?) homotopy (co)limits commute with homotopy (co)limits, so we can get the spectral sequence of a homotopy (co)limit in this language. Also mapping spaces commute appropriately, so we can get the AH-SS.

Step two really is the input to this entire game. Note that $\alpha$ commutes with finite homotopy (co)limits (i.e. cones) so this is reasonable.

Step three is the spectral sequence of a filtered homotopy type. There is always a spectral sequence, but it may or may not converge to the groups of the homotopy (co)limit (depending on exactness properties of homotopy (co)limits on $A_2$).

I think many spectral sequences can be looked at in this way (certainly Grothendieck, AH, homotopy limit/colimit; probably also Adams).

To expand on Ben Wieland's example of the AH-SS, in this case $D_1 = D_2 = SH,$ $\pi_1 = \pi_2 = \pi$ is the stable homotopy groups, corresponding to the natural t structure, with heart abelian groups, $X$ is a fixed spectrum, $\alpha(E) = Map(X, E).$ Then $\alpha$ sends homotopy limits to homotopy limits, so we should use the filtration corresponding to $\pi,$ writing $E$ as the homotopy limit of its Postnikov filtration. The subquotients are Eilenberg-Maclane spectra corresponding to the homotopy groups of $E,$ and so we get a spectral sequence

$$ E_2^{*,*} = \pi_*\alpha(\text{subquotients of }E_*) = \pi_* Map(X, H\pi_*E) = H^*(X, \pi_*E) \Rightarrow \pi_*\alpha(E) = H^*(X, E)$$

which is what we wanted.