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I imagine Everstones would be obscenely popular so people who have the means to take care of, say, a magikarp don't have to worry about it suddenly turning into a 20 foot long rampaging sea serpent.
I'd probably go with a humanoid type like Mr. Mime or Hypno, something roughly the same size as a person that has opposable thumbs. I feel like it would be easier to communicate with, understand, and train something that's a closer approximation to a human being than other Pokemon, and a human-like type would have a much easier job navigating a world dominated by human civilization (no pawing at the door to be let in, they can just use the dang doorknob).
On the other hand, they probably tend to be far more emotionally demanding than, say, a herdier, as well as being more intelligent on average, so it probably wouldn't be as easy as raising something like a sunflora that can just stand out in your yard for hours at a time to get its daily nutritional intake. Also, owning a Hypno would probably be a good way to be put on a watchlist.