You'd probably be better served by an array for this, where you could just do:
$array[int(rand(scalar(@array)))];
to choose a random element.
You can do much the same thing, though, by pulling the hash keys into a list and choosing one randomly:
$hk = (keys %hash)[int(rand(scalar(keys %hash)))];
$hv = $hash{$hk};
If you'll do it often, pull the keys into an array and just use that:
@arr = keys %hash;
$hk = $arr[int(rand(scalar(@arr)))];
$hv = $hash{$hk};
With both of the hash solutions, you end up with two copies of all the keys, so it's less memory-efficient than an array, but probably fine if you have a few thousand entries or less.