What is the significance of having Prime Ideal Theorem in models for failure of Axiom of Choice? Prime Ideal Theorem says:

PIT: Every ideal on a Boolean algebra can be extended to a prime ideal.

It follows from Axiom of Choice but is weaker than it.
In many cases I saw that people check for having $\sf PIT$ in models for $\sf \neg AC$. 

Why? What is the significance of having Prime Ideal Theorem in the absence of $\sf AC$? What sort of choice related set theoretic tools are available merely by $\sf PIT$?  

 A: This is kind of tangential to the OP, but it's too long for a comment.
First, let me point out that the general question of which choice principles imply which others is interesting from a purely technical standpoint. The proof techniques required to show that one form of choice does not imply another are often quite subtle.
This doesn't address your question, though, which I see as having two parts:

(1) What tools does PIT provide us?

which is your main question, but also 

(2) Why are we interested in models of ZF+PIT+$\neg$AC?

For (1), PIT has powerful applications in logic and point-set topology, in the form of various compactness results: e.g., the compactness theorem for first-order logic in arbitrary languages, and the Hahn-Banach theorem in topology. Of course, it doesn't get us everything - Tychonoff's theorem, for example, is equivalent to full AC - but it gets us a fair amount.
For (2), even if what we really care about is ZFC, we often find that questions about the universe are closely tied to properties of inner models - that is, transitive proper classes containing all the ordinals which are models of ZF. These include, for example, $L(\mathbb{R})$, the smallest inner model containing every real. In many cases of interest, AC fails in inner models: for example, in the presence of large cardinals, $L(\mathbb{R})$ satisfies the axiom of determinacy, a very different sort of animal. It suddenly becomes important to know what choice principles might hold in such inner models, and this is tied up with understanding what choice principles imply what other choice principles. For the specific case of PIT, this is admittedly purely speculative on my part - I don't know how important this specific form of choice is in the context of inner models; for example, the axiom of determinacy destroys it, so it tends not to hold in $L(\mathbb{R})$ - but I do find it a good heuristic argument for caring about separations of choice principles in general.
