It depends on both the image and how much editing you are planning to do. Let's say you're only going to adjust levels. Then

8-bit JPEG -> 16 bit PSD -> levels adjustment -> 8-bit JPEG

Will give identical results to:

8-bit JPEG -> levels adjustment

This is because an 8-bit image converted to 16-bit is identical to the 8 bit version (the lower order bits will all be zero) and there's no difference between doing the levels on a 8 bit image (which will be computed internally at a higher precision and then rounded) and doing the levels command on a 16 bit image and then converting to lower precision.

However, if you do several steps, then keeping a high precision intermediate version of the image will give you better results. If you keep rounding to 8-bit after every single operation, you're very quickly going to lose tonal range and end up with a heavily posterized image.

Now I said it depends on the image content, a very contrasty image with posterization will generally go unnoticed, whereas a low contrast image with subtle graduations in the sky or in skintones will be heavily degraded by posterization.