Result looks like: 

You need to understand that the draw() method is executed approximately 60 times a second (frames), this means that it is best to draw the whole display inside the loop rather than build up a picture over several frames. If you do that then you have to clear the display at the beginning of every frame with the background command. In Processing the draw() method is essential if you want to detct mouse movement and key presses or any other user input.

So it is good practice to initialise all your variables in setup because this is executed first and only once. The draw method can be used to draw the display based on changes in the variables such as mouseX and mouseY

So I have modified your program to take these factors into account and here is the code.

1. //global variables
2. int distFromEdge = 100; //distance from the edge of the window
3. int startYear = 1997; //first year
4. int endYear = 2012; //last year
5. int currentYear = startYear; //current year in the loop; start at startyear
6. int yearSpacing = (endYear-startYear); //how many spaces are there?
7. int circleSize = 30; //node diameter
8. float x = distFromEdge; // X-Position of node for the loop
9. int circleY = height/2;  // Y-Position of nodes always - must be along the line
10. float[] circleX = new float[yearSpacing+1]; //an array of each nodes x position
11. int indexYear = 0; //this will be the index for circleX[] above
12. int intervals;
13. boolean circleOver = false;//are we over a node?
14. 
    
15. void setup() {
16.   size(1080, 240);
17.   smooth();
18.   background(151, 151, 2);
19.   fill(255);
20.   //circle pos
21.   ellipseMode(CENTER);
22.   // Calculate the distance between year nodes / circles
23.   intervals = (width-(2*distFromEdge))/yearSpacing;
    
24.   // Calculate all the circle positions
25.   for (int i = 0 ; i < yearSpacing + 1; i++)
26.     circleX[i] = distFromEdge + intervals * i;
27. }
28. 
    
29. void draw() {
30.   // Draw the horizontal line
31.   stroke(0);
32.   strokeWeight(10);
33.   strokeCap(ROUND);
34.   line(0+distFromEdge, height/2, width-distFromEdge, height/2);
35.   // Draw the circles
36.   strokeWeight(6);
37.   for (int i = 0; i < circleX.length; i++) {
38.     // What size circle to draw
39.     int csize = circleSize;
40.     if (i%5 != 0)
41.       csize = circleSize/2;
42.     //
43.     if (overCircle(circleX[i], height/2, csize)) {
44.       stroke(255, 206, 0); //change color
45.     }
46.     else {
47.       stroke(0); //or stay black
48.     }
49.     ellipse(circleX[i], height/2, csize, csize);
50.   }
51. }
52. 
    
53. boolean overCircle(float x, int y, int diameter) {
54.   float disX = x - mouseX;
55.   float disY = y - mouseY;
56.   if (sqrt(sq(disX) + sq(disY)) < diameter/2 ) {
57.     return true;
58.   }
59.   else {
60.     return false;
61.   }
62. }

Brilliant, and so much simpler than what I was attempting. So you calced all the circle positions in setup() - and moved all the actual drawing to  draw() ;). 

Added text labels for each year: