In Avengers v X-Men #1 Cyclops seems to think that Phoenix will somehow counteract the destruction of all but 200 mutants (from some earlier story I guess). Is this based off an existing known Phoenix power or usage, or is he just guessing or hoping?
As mentioned by jonnyflash, it seems reasonable to assume the Phoenix could, if it wished, restore the mutant population. The Phoenix Force is one of the most powerful forces in the Marvel universe and has sufficient power to accomplish whatever it chooses. For example, it has previously reached across time to send a telepathic command and alter the timeline. (See Morrison's New X-Men.)
Then the question becomes, why does Cyclops think it will help mutantkind rather than just scour the planet of life?
This is still unclear, but there are a few pieces that can help us see this from Cyclops' perspective.
Previous hosts of the Phoenix Force have shown control over its abilities. Since Hope is the next vessel of the Phoenix, it seems likely she will be able to exert some influence over what it does. Cyclops, I imagine, is gambling that Hope will be able to influence the Phoenix enough to use it to restore the mutant race.
The Phoenix Force is a force of death, but also rebirth. The symbol of the phoenix represents this. Jean Grey, when host to the Phoenix, referred to its purpose as "to burn away the obsolete". Presumably, Cyclops thinks that mutants are not obsolete, since he sees mutantkind as the future of humanity. (One hopes he believes humanity will also be spared the judgment of the Phoenix, although this may be a secondary concern.)
Jean Grey may continue to influence the Phoenix. This is fairly speculative, but it's the sort of thing Cyclops may actually believe due to his extreme faith in Grey and her abilities. The Phoenix Force has repeatedly shown an attachment to Jean Grey and mutantkind, so there might be something to this.
Short answer: It's probably a guess, but there's some interesting evidence. He probably thinks Hope will be able to control the Phoenix, and will use its power to safely restore the mutant population to its previous levels of growth.
Hope's birth was viewed by the remaining mutant population as the birth of the 'Mutant Messiah'. So, if she becomes the Phoenix, she may restore the mutant population to what it was before its decimation. I say if Hope becomes the Phoenix because she isn't the only potential avatar of the Phoenix Force. Rachel Grey/Summers and Emma Frost have been the host of the Phoenix before. Not to mention Kid Omega, who was shown as a future Phoenix by Deathlok's vision.
As a "cosmic being" in the Marvel Universe, there is very little the Phoenix cannot do. In the past it has been limited only by the ability of its host to use its power.
As I understand the issue and the phoenix legend, Cyclops is hoping that the regeneration that brought by the destruction will help restore all the Mutants who lost their power or return the ability for the X-gene to flourish. I have a feeling this will be explained more in later issues.