This was a widely reported problem on this forum back when the G4 was a current model. It was traced down to a limitation relating to a disclaimer in the mini's spec about working with "non-coherent" displays. If you search the page from the link below using the search word "coherent" you will find the disclaimer.
http://support.apple.com/kb/sp65
The solution that worked for most people was to drop the pixel clock frequency. ATSC 1080p60 timing has a pixel clock of 148.5 MHz. You can use a program called SwitchResX and use it's built-in function for building custom timings using CVT-RB. The RB stands for reduced blanking and it causes the pixel clock to drop to around 138 MHz IIRC. For most people that was lower enough that it did the trick.
The first thing you will want to do the next time you start the mini hooked up to that TV is to lower the resolution to something like 720p before the picture goes out. The pixel clock is much lower for 720p so the picture won't blank out on you and you'll be able to use it long enough to do everything you need to do with SwitchResX.
Write back if you need more help or further explanation.