There is a tutorial for this effect here. It is for PS but you can still achieve the same look in PS. Here are the basic steps below:

1. Add extra canvas to your photo: Image > Canvas Size add some extra here. The amount is not really important as you will crop at the end.

2. Add Black Layer on top, fill with black.

3. Add Noise. Filter > Noise > Add Noise. Use Uniform set around 150 percent with Monochromatic Checked

4. Add Gaussian Blur. Filter > Blur > Gaussian Blur set to around .5 pixels

5. Apply Motion Blur Filter. Filter > Blur > Motion Blur. Use Angle of 60 (change for different angle) Distance 50 Pixels.

6. Change the blend mode to Screen on the layer with your "rain".

7. Duplicate Your Noise layer and Hide the layer. (So you have it for later use if you want to re-do the effect.)

8. I do not believe PS Elements does not have adjustments layers (way the tutorial recommends). So you will go Image > Adjustments > Levels . Drag the Black and White Arrows inward till you get the effect you would like. Photo Below:

9. Crop away extra canvas.

Now you have rain!

On another note you can find decent rain (or snow) brushes for Photoshop (CS X or Elements). Here is one set I have used and a Google search turns up many results.