public static int getRandom(int[] array) {
    int rnd = new Random().nextInt(array.length);
    return array[rnd];
}

Just store the index generated in a variable, and then access the array using this varaible:

int idx = new Random().nextInt(fruits.length);
String random = (fruits[idx]);

P.S. I usually don't like generating new Random object per randoization - I prefer using a single Random in the program - and re-use it. It allows me to easily reproduce a problematic sequence if I later find any bug in the program.

According to this approach, I will have some variable Random r somewhere, and I will just use:

int idx = r.nextInt(fruits.length)

However, your approach is OK as well, but you might have hard time reproducing a specific sequence if you need to later on.

Use the Random.nextInt(int) method:

final String[] proper_noun = {"Fred", "Jane", "Richard Nixon", "Miss America"};
Random random = new Random();
int index = random.nextInt(proper_noun.length);
System.out.println(proper_noun[index]);

This code is not completely safe: one time out of four it'll choose Richard Nixon.

To quote a documentation Random.nextInt(int):

Returns a pseudorandom, uniformly distributed int value between 0 (inclusive) and the specified value (exclusive)

In your case passing an array length to the nextInt will do the trick - you'll get the random array index in the range [0; your_array.length)