Given empty list, then adds String (ArrayList implementation)

Your implementation has a few errors.

public StringArrayList(int initialCapacity){
    strings = new String[0];
}

This type of constructor is meant to allocate with the initial capacity, not with "0".

And most importantly, your add method doesn't actually add a new element, but overwrites your current internal array, and places a new element at the 0th index (without incrementing the size variable).

public boolean add(String content){
    boolean add = false; //unnecessary, can replace bottom `add` with `false` for same result
    strings = new String[10]; //overwrites internal array of "list" with a new array
    for(int i = 0; i < strings.length; i++){
        if(strings[i] == null) { //will always replace only first element as this is a new array
            strings[i] = content;
            return true;
        }
    }
    return add;
}

The size variable is meant to keep track of how many elements the current array contains (which would be incremented when you add a new element).

Knowing that, you'd be able to add a new element instrings[size] to add a new element, in case size < strings.length was true.

If that condition is not met (you want to add more than the current capacity), it should "resize" the internal array by creating a new array, copying all previous elements into the new array, then replacing the old array, and adding the new element to the new array.

EDIT: Your new code

public boolean add(String content) { 
    strings = new String[strings.length+1]; //still erases the internal array!
    strings[0] = content; //only modifies the first element, rather than add to the list
    size = 1; //if it was implemented correctly, this would be size++;
    return true; 
} 

Make a test that adds TWO elements to the list, and assert if the list contains both of them.

EDIT2:

Here is a solution that ought to work:

public class StringArrayList {
    private int size = 0;
    private String[] strings;

    // Constructors
    public StringArrayList() {
        this(0);
    };
    public StringArrayList(int initialCapacity){
        String[] innerStrings = new String[initialCapacity];
        this(innerStrings);
    }
    public StringArrayList(String[] strings){
        this.strings = strings;
    }

    public int size() {
        return size;
    }

    public String get(int index){
        return strings[index];
    }

    public boolean add(String content){
        if(size == strings.length) {
            String[] newStrings = new String[size+10];
            for(int i = 0; i < size; i++) {
                newStrings[i] = strings[i];
            }
            strings = newStrings;
        }
        strings[size++] = content;
        return true;
    }
}

EDIT3:

public boolean add(String content){ 
    if(size == strings.length) { 
        String[] temp_list = new String[strings.length]; //no need to allocate a new array here if this is just to store your current array
        temp_list = strings; 
        strings = new String[size++]; //wrong, this allocates a `size`-long array and increases `size` by 1 afterwards, rather than create a new, larger internal array
        strings = temp_list; //this just overwrites your internal array with the old array which is not increased in size

     } 
     for(int i = 0; i < strings.length; i++) { 
         if(strings[i] == null) { //this is not necessary with proper `size`
             strings[i] = content; 
         } 
     } 
     return true; 
}