I am looking for examples where choosing one possibility out of many for a metric gives nice topological properties compared to the other choices.
Nice is defined as compact, Hausdorff, or whatever the reader prefers. For example, in $\mathbb{R}^n$ with the Eucledian metric we have the Heine-Borel theorem.
In general this doesn't exactly make sense. Nice purely topological properties like compactness and Hausdorffness are dependent on topology, not metric (Actually metrics always give Hausdorff topologies). And the usual sensible meaning of "choosing one of many metrics" requires they give the same topology. Otherwise you're just asking whether different topologies give different properties.
Now, this isn't to say it's totally meaningless. In functional analysis for example we may have a single large underlying space that makes sense in multiple contexts, and thus introducing different metrics, or more generally different topologies may be of interest. Or we could talk about giving the rationals the usual or p-adic metrics, where we're considering the metric as coming from an absolute value that interacts in "nice" ways with the algebraic structure of the rationals.
And in the general metric space context, we may ask if there are two metrics on the same space such that a property of metric spaces (as distinct from a topological property like connectedness, compactness, Hausdorffness) changes, but such that the collection of open sets are the same. Two examples: One is fairly canonical. For any metric space $(X,d)$ there is a new metric defined:
$$d'(x,y) = \frac{d(x,y)}{1+d(x,y)}$$
equivalent to $d$ such that the new space is bounded. More interestingly: there is a metric, though I forget the construction, that can be put on the irrationals that (a) gives the same open sets but (b) makes $\mathbb{R}\backslash\mathbb{Q}$ complete as a metric space.
To round out this answer, just in case you don't know:
Properties that are topological (that is, depend only on the open set structure): connectedness, compactness, separablity, first/second countable, Hausdorff, normal, regular, continuity of functions, in some sense convergence of a sequence.
Properties that depend on metric: boundedness, completeness, uniform continuity, Cauchy-ness of a sequence.
