Having enough information to specify the set isn't enough to pick out one particular member.

For example, if i say "the colours teal, maroon, and taupe", you have enough information to know what's in the set, but no extra information that would let you pick one element out of it.

Well the issue is whether its possible in principle to define such a function for all possible sets. But if you can enumerate the elements of the set, such a function is simply to pick the first element.

So the issue is how much information is necessary to guarantee that its possible in principle to construct such a function. It should be clear that enough information to enumerate each set is enough to create a decision function. The remaining question seems to be whether one can specify a countable set of countable sets without specifying enough info about each member set to enumerate the set's members. It seems trivial that a set with countable elements is enumerable, but I might be missing some subtlety.

You just listed the elements, that's the easy case. You can pick one of the listed ones, like Teal.