It is possible, but you will not get the best quality. The Kinect won't give you very good geometry data to work with, i.e. when you go to print it out it won't resemble your grandson. If you scan with colors and print in full-color sandstone or full-color plastic you'll get something better, but still not great. Whether you're scanning just geometry or geometry and texture, there is a huge amount of post-processing you would have to do just to clean up the data before printing like removing artifacts that have generated extra geometry that isn't supposed to be there. If you are planning on printing in color, you would also probably have to do a lot of texture correction because from what I've seen most software programs don't do a very good job of re-projecting the colors back on. You would also need to do color correction because most scanners don't have great lenses or cameras for capturing good color.
If possible, I would recommend getting your grandson scanned in a photogrammetry rig if you live close to a place that does 3D human scanning and then ask for the file. I recommend a photogrammetry-based scanning solution because it is nearly instantaneous (great for small children who tend to move around a lot) but will give you decent results.
If you're near Palo Alto, California you can also visit the Artec 3D Showroom on University Ave and get him scanned there (full disclosure: I used to work for Artec). The scanning takes 12 seconds and is slower than photogrammetry but will give you much better results because it uses four of Artec's industrial Eva scanners that are meant to capture geometry in high detail. They are white-light-based structured-light scanners that are okay to use with a person's eyes open, unlike laser scanners.
Here is the print /u/G33kinator did of my scan from Artec's Shapify Booth. There are also Shapify Booths in Europe, but most are in the UK. Artec also has [Viewshape.com](viewshape.com) where people upload models of things they have scanned using Kinect, Asus Xtion and Carmine (other PrimeSense scanners), and Artec scanners and edited using Artec Studio. This will give you the best idea of what to expect when scanning with each of the different scanners.
In any case, there are best practices for 3D scanning people such as holding as still as possible and not moving the eyes when scanning. Shiny, translucent, and thin materials like glasses or loose sheer fabric should not be worn during scanning because most scanners won't pick it up and will result in a messy blob. It's also a good idea to tilt the head slightly upward when scanning. A lot of 3D-printed people end up as desk accessories or mantelpiece decorations and looking straight forward makes the person look very zombie-like, especially if the person isn't smiling.
Another option would be to print a wax mold and cast in metal or order a metal print from somewhere like Sculpteo or Shapeways. Getting it cast in metal will reduce the number of visible imperfections in the model and looks great on a desk.
tl;dr 3D scanning at the consumer level just isn't there yet because of price and technology. Get your grandson scanned by a professional and print the model.