I would decide by running some recording tests, and deciding which image you actually prefer - the on-board stabilisation, or doing it in post on the PC. There will be pros and cons both ways - although I wouldn't worry about things like CPU overhead, unless you've actually read somewhere of this being an issue?
This article gives a little bit of a run down of the stabilisation feature, if you haven't seen it.
Based on the article, it looks like the stabilisation is being done in the sensor. In a lot of cameras, the full sensor isn't used for recording, certainly not in all recording modes. For stabilisation, these cameras sample a larger area of sensor elements than what is being recorded, work out where your image has moved between frames, and record the area/window they consider to be your intended image, smoothing out your movements.
The article notes that you can't record a full resolution image in some modes when using the stabilisation, so it's almost certainly sampling the full sensor, and then stabilising by reducing the number of sensor elements being recorded, changing where the recording window is, based on your movement. Note with this method that the camera might not always stabilise in a way you consider to be "correct", and you'd have no control over that.
Stabilising with the PC will give you more control over what is done, but the PC software will also be interpolating (educated guessing) where required to create extra image data for your stabilised image. If you don't need the highest camera resolution, the stabilised visual data from the GoPro might be preferable to stabilisation corrections generated in PC software. But if you want to record in the highest resolutions, you'll be cropping images from the GoPro to do that, which may not be ideal - taking the unstabilised GoPro image and stabilising on a PC may be the better option for some resolutions.
Like I said, I would test both, and see what you think of the results for yourself. It's useful to know how both stabilisation techniques work, and other than that there's no correct answer for everyone - it will depend on exactly what you're recording, what quality it needs to be recorded in, and which method produces the best looking result to you when completed. If you don't need full resolution, and you're happy with how the stabilised GoPro image looks, I'd probably go with the camera stabilisation - but it was certainly noted in the article above that in some cases the stabilised image didn't look that great, so I'd definitely test it on what you're doing first.