I personally wouldn’t bother - the Radeon 4000 series is thoroughly obsolete and will only be a minor bump compared to the Intel GMA HD board (even if it’s still terrible). You will need to find the 512MB board if your heart is absolutely set on replacing the motherboard (at the cost of a complete identical system), since the GPU VRAM is soldered like the GPU. In addition, you will need to replace the heatsink since your Intel GMA system has a different style heatsink. The benefit of getting away from the Intel UMA architecture isn’t worth the cost to replace the board in your case.
You’re better off investing the money you’d sink into this minor upgrade into a used Haswell or Skylane laptop that’s significantly better, and will also support modern protocols like USB3 and modern components like DDR3L/DDR4 memory and M.2 AHCI SSDs. It doesn’t make sense to scrap this - keep it as a backup system until it dies if you buy a modern notebook to replace it. Haswell has endured surprisingly well, especially with the M/MQ systems; I usually don’t hold onto hardware for 7 years like my E6540, but Haswell has endured so well I made an exception (plus I have a Win7 system, so that’s partially why). I’m planning on holding out until I can get a Ryzen 3000 laptop to replace it, especially if I can get one with TB3.
The only caveat is most Haswell systems seem to run Windows 7 (especially business models like my Latitude), so you will probably want to factor the cost of a Windows 10 Pro/Home license into your total cost unless you can still get the free upgrade with your product key.