[Lost Mine of Phandelver spoiler, proceed with caution]
At page 41 from the "Lost Mine of Phandelver", last paragraph, it reads:
> Whether the characters sought the information from Agatha or Reidoth, negotiated with Hamun Kost, (...) they now know the location of Wave Echo Cave. (...)
But on page 29, describing the Old Owl Well "quest" where the chars meet Hamun Kost, there's no information regarding the mage Kost knowing where the Wave Echo Cave is. Am I missing something?
I know I can create that on my own, but now I'm just trying to stick with the text book and understand the campaign as it was originally designed.
Thanks in advance! 
Well, presumably he does, though it isn't explicit. It says that while he won't explain why he is in the region (nor does the book explain why he is in the region), he is "willing to provide information the party needs if it does a favor for him." I'm not sure what other information that would be, and the bit on page 41 that you quoted seems to be dispositive here.
Of course, that raises further questions -- how does he know? Are the Red Wizards in general aware of the location? Do they have any kind of relationship with the Black Spider? And why is Kost there, and why does he need to know that the Terraseer built the tower at the well? Presumably he is searching for Netherese treasure, and perhaps he needs to know the name in order to access something within... but the point is, you will have to fill in information on your own unless your players are like mine and take the 'kill first, ask questions never' approach (and even then it still is worth it to ask questions like you are doing since you never know what players will do and its good to know reasons and relationships).
Thanks a lot, @paulstrait. The "willing to provide information the party needs" really answers my question -- it's up to me to imagine how the heck he knows that.
We used to play some old tabletop RPGs, hence they are more inclined to try to fix the problem negotiating and discussing instead of shooting an arrow first... The problem with that is it becomes really hard (for the DM) to predict what the players will do. 
I've been playing with most of the guys in my group for 10 years, and we all had different experience before that. No amount of past experience, creativity, good sense, etc. seems to have much of an effect.
(That said, one of the players who might usually be inclined to think of alternatives to combat is playing a barbarian, and indeed in this campaign has been actively foiling attempts by the other PCs to take a less blunt approach. For example, the party decided to wear the Redbrand disguises they found, but his character didn't 'get' that this was for the purpose of disguise--hilarity ensued when the rest of the characters realized that the barbarian fundamentally did not understand/agree with the tactic they were attempting.)
I think he should be able to perform some foul Thayan divinecromantic ritual that would allow him to discern the location they seek, perhaps by examining entrails for foresight along the adventurers' path or augury that requires shared bloodletting to sift through ancestral memory for answers.
But he only does it if he gets an answer about which mage built the tower at Old Owl Well. This will be especially interesting if the PCs have made a deal with Sister Graele - whose question will they ask of Agatha? What's more important - the good will of the Harpers or the location of Wave Echo Cave?
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