But doesn't this just mean it could be pretty much any point?
Yes, of course. But if the space is a Hausdorff space (almost all interesting spaces are), after you've selected such a point, I just select another, smaller neighbourhood that doesn't contain that point you chose (in a Hausdorff space I always can do that), and you will have to give me a point in that neighbourhood as well. Indeed, I can repeat that process infinitely often, and since each time you'll be forced to present me yet another point, the end result is that there will have to be infinitely many points in any neighbourhood.
Not necessarily anywhere near a boundary/limit of S?
A limit point doesn't need to be near a boundary. Indeed, for $\mathbb R$ with the usual topology, every point is a limit point of the full set $\mathbb R$, and that set doesn't even have a boundary (or more exactly, its boundary is the empty set)!
Now on the distance of the chosen point from the limit point, of course if you even want to speak of "near the limit", you have to have a concept of nearness; usually in the form of a metric. And in that case, I can play essentially the same game as above, except this time taking the distance into account:
I start by providing some neighbourhood, and you give me some point in the neighbourhood. Given that we are now in a metric space, that point will have a certain positive distance from the limit point. Thus I present you in the next step the neighbourhood that only consists of those points that are closer than half that distance. So now you'll have to provide me with another point in that neighbourhood, which means especially a point that is less than half as far away from the limit point as the point you've originally chosen. Again, I can repeat that infinitely often, and unless you run out of points to give me (in which case the point wasn't a limit point to begin with), you'll end up with a sequence of points that come arbitrary close to the limit point (the sequence of distances clearly converges to zero).
On the other hand, if you don't have such nice properties like Hausdorff or even metric, then you might indeed get away with choosing just one point. For example, take any set with the trivial topology. Then whatever point you chose, there's only one neighbourhood: The complete space. So indeed, in that space, all you need for a point to be a limit point of a set is to have one other point that belongs to the set.