My suggested answer assumes that the stress ball is hard and not easily squeezed out of shape - although a little distortion won't matter too much.
Do you have a measuring jug in your kitchen? You know, the sort of glass or plastic jug that has a scale on one side (in cubic centimeters, say), that you can measure the volume of a liquid that is in the jug?
If so, you can use it to answer your Q. Like this . . .
Fill the jug up to about half way with water and note the reading on the scale. Now put the stress ball in. If it floats, gently push it just beneath the surface of the water with a pin. Make a note of the new reading on the scale of the jug. The difference between the two readings = the volume of the stress ball.
You can calculate the surface area from:
1. Volume, V = (4/3).pi.r^3,
2. Surface Area, A = 4.pi.r²
where r= radius.
So: from (1) r^3 = 3V/4.pi, so r can be calculated.
Then put this value of r into (2) to calculate Surface Area, A.
Another idea: have you got a spare tennis ball, a knitting needle and a little paint?
If you carefully paint a thin line around the tennis ball, push the needle right through the centre of the ball and holding the ends of the needle, carefully roll the tennis ball (with the paint wet) along a piece of paper, (being careful not to 'over-roll' the ball), you can thus find its circumference.
From that, you can find the radius.
Not such an accurate method as the first - but it just might give you some other - and better - ideas!
Me: I'm an engineer. I'm used to solving 'everyday' practical problems!