Suppose I have a poset $P$. A subset $Q$ of the elements of $P$ has the property that, for any two elements $a,b \in Q$, $a$ is "connected to" $b$ through a chain of comparisons with the elements of $Q$. Hence, e.g., $$a > c > d > b$$ or $$a < c > d < e < b,$$ etc., with $c,d,e \in Q$.
I've been calling $Q$ a "topologically connected pocket in $P$", but I've been told that this expression is confusing.
I'm really not all that familiar with poset theory. Is there a more formal name for a subset with this property?
Every poset is a directed graph with a directed edge $a \to b$ whenever $a \le b$, and in a directed graph the concept you're looking for is called being weakly connected (as opposed to "connected," which is reserved for directed paths). If $Q$ is maximal with respect to this property then it is a weakly connected component. I think that term would be fine here although you might have to define it.
The corresponding concept also makes sense in a category; in this context the chain of comparisons you're describing is called a zigzag which may be useful terminology. Zigzags are morphisms in the localization of a category, where we formally invert some morphisms, and the operation here can be thought of as formally inverting all the morphisms in a poset regarded as a category. Of course you don't need to know any of this; this is just background to show that this is an important and relevant concept and not an ad hoc construction.