Is there any meaning in marginalizing over the conditioned variable? If we have P(A|B), does the marginalization 
$\sum_{b \in B} p(A|B)$
have any meaning? How is this called?
 A: The marginalization you are asking for does not have any interpretable or useful meaning. For example, let's look at a medical example. 
Observe the conditional table for the probability of a positive xray reading given the presence of cancer. The expression for this table may be written as $Pr(X|C)$
X \ C True  False 
+     0.9   0.2
-     0.1   0.8

The expression you are asking about would be: $\sum_{c \in C} Pr(X|C)$. 
The table equivalent to this expression would be,
X   Pr(X)
+    1.1
-    0.9

Since the numerical values no longer sum to one, they have lost their meaning. We can use the following expression to get interpretable results, $Pr(X) = \sum_{c \in C} Pr(X|C=c) Pr(C=c)$. 
Of course to do the calculation, we will need a table of the probability of cancer.
C      Pr(C)
True   0.05
False  0.95

Now, we can compute $Pr(X) = \sum_{c \in C} Pr(X|C=c) Pr(C=c)$
X   Pr(X)
+    0.235
-    0.765

This term now properly described the probability of positive vs negative xray results without being conditioned upon cancer.
Note: These numerical values are not accurate. They were created for the sake of example.
