Videomaker – Learn video production and editing, camera reviews › Forums › Technique › Editing › any recommendations? › This will be an uphill battle
This will be an uphill battle: you can't very much improve what isn't there. First, I would find the best shot(s) of your child and make some stills. Try using the tools in Photoshop, PaintShop Pro, etc., to enhance the still. Most helpful tools will probably be clarify, curves and unsharpen mask. Brightness and contrast will probably do little more than increase the noise in the image but give them a try anyway.
Working with the video in your NLE I would first apply levels, then curves and if necessary secondary color correction, then unsharpen mask. Work with your monitors to bring the video up to the best quality you can get. It probably won't be too good but perhaps better than it is now.
As for the audio, there's probably very little you can do at all. Perhaps some noise reduction to try to isolate the voice from the background and perhaps work with EQ to see if adding a bit of high end will pop the voice a little.
In all of this, less is more; don't apply too much of any tool. And watch the monitors!