Why wasn't Jiraiya brought back with Edo Tensei? Was it ever explained why Jiraiya wasn't brought back via Edo Tensei when just about every other powerful ninja in history was?  

From my understanding, the summoner only needs a good bit of genetic material to bring someone back and Jiraiya's body was intact when he died.
The point is quite simple.


   In order to perform Edo Tensei you need a sample of DNA of the deceased. If you remember how Jiraiya died, he's been in the depth of the water and it's not feasible to recover him. 


And indeed in this Wiki on Edo Tensei, it is stated that  


   Kabuto wanted to reincarnate him, but his "body lies at the bottom of the ocean where the water pressure is too great to reach. However, he claimed that the bloodstains on the Six Paths of Pain's weapons may provide enough DNA."

 He wanted to reincarnate Shisui Uchiha too, but he "could not find his body, though he suggested that Shisui's crushed eye, which is in Tobi's possession, may be sufficient. Later, after learning that Shisui's other eye was on the battlefield, he seemed excited at the prospect of obtaining it. However, Itachi incinerated it."  


If you're reading the Manga, the fact about Jiraiya is stated in the Volume 55, Chapter 520, page 14, so you can read if you got there:


    Kabuto: Unfortunately, I had no luck finding Uchiha Shisui's corpse, and Jiraiya's body rests deep in the ocean, below unfathomable depths.  Though Danzou's right eye and the Six Paths of Pain's weapons should still carry their DNA...


So, that's a pretty simple but fundamental requirement.