I think I might be one of the only people in the US that has BUILT an HDMI cable... so here goes...
There are retermination kits available online, which generally will accommodate a 26-28AWG wire. However, if you are trying to reterminate a thicker cable, it's not going to work. The reason being the solder cups/joints are far too close to one another to use a larger gage wire, and not have anything ground out.
So, the catch to all this, is that while you can build shorter HDMI cables from a thin cable stock, why wouldn't you just buy a new one? The max length of HDMI you could build with a 26-28AWG cable, would be around 30 feet (if that). So it doesn't really make sense to build a new cable, when you could get a replacement and avoid all the hassle. And trust me, it's a HUGE hassle.
Hope this helps.