Make a sketch. The Earth is a sphere (not flat) - so, draw a circle representing the spherical Earth
of the radius  r = 3960 miles.

On the top of the circle, draw a person h = 6 ft =  miles tall, standing on the Earth.


From the level of the person's head, draw a tangent line to the Earth.
This tangent line will represent how far this person sees the surface of the ocean.
Also, draw the radius of the Earth to the tangent point. 


On the sketch, you will see the right-angled triangle with the hypotenuse length of

      r + h = 3960 +  miles

and one leg of the length  r = 3960 miles.


So, you can find the other leg of this triangle, which is a tangent to the Earth surface, 
from the Pythagorean equation

   d^2 + r^2 = (r + h)^2.


Here "d" is the distance to the horizon, in miles.


So, your friend is right.


Calculations give  d =  = 3 miles.