I can think of two solutions, but there could be more as I'm not very good at coding in general yet.

First of all, you can use perlin noise. With the code

var noiseSeeds = [];
//This changes the noise value over time
var noiseTime = 0;
var x = 0;
var y = 0;

function setup() {
    createCanvas(400, 400);
    //This will help for making two separate noise values later
    noiseSeeds = [random(100), random(100)];
}

function draw() {
    //Finding the x value
    noiseSeed(noiseSeeds[0]);
    x = noise(noiseTime)*400;
    //Finding the y value
    noiseSeed(noiseSeeds[1]);
    y = noise(noiseTime)*400;
    //Increasing the noise Time so the next value is slightly different
    noiseTime += 0.01;
    
    //Draw the point
    stroke(0);
    strokeWeight(10);
    point(x, y);
}

You can create a scribble on screen. You would have to use createGraphics()in some way to make this more efficient. This method isn't the best because the values are generally closer to the center.

The second solution is to make a point that has two states - far away from an edge and close to an edge. While it is far away, the point would keep going in relatively the same direction with small velocity changes. However, the closer the point gets to the edges, the (exponentially) bigger the velocity changes so that the point curves away from the edge. I don't know exactly how you could implement this, but it could work.