You can use a boolean. Then you can change the color once they touch instead of while they touch. They have to first be apart again before the color can be changed again. For this to work, it's also good to use a variable to store the background color.

Adapted Code

1. float diam = 100;
2. boolean currentlyTouching;
3. color bgColor = color(255);
4.  
5. void setup() {
6.   size(500, 400);
7.   strokeWeight(5);
8.   noFill();
9.   smooth();
10. }
11.  
12. void draw() {
13.   background(bgColor);
14.  
15.   float x = width/2;
16.   float y = height/2;
17.  
18.   float d = dist(x, y, mouseX, mouseY);
19.  
20.   if (d < diam) {
21.     if (currentlyTouching == false) {
22.       bgColor = color(random(255), random(255), random(255));
23.     }
24.     currentlyTouching = true;
25.   } else {
26.     currentlyTouching = false;
27.   }
28.  
29.   stroke(255, 255, 0);
30.   ellipse(x, y, diam, diam);
31.  
32.   stroke(0, 255, 0);
33.   ellipse(mouseX, mouseY, diam, diam);
34. }

Something like this?