I would first urge caution that Start-Process may not be exactly what you desire. This will open a new process in a new command window (cmd).
I would encourage you to read the blog post PowerShell and external commands done right FYI.
However, there are a few ways to escape the quotes. One way to include double quotes in a string is to use a single quote to encapsulate the whole string for example:
$String = 'video="Integrated Camera"'
Alternatively you can escape with a backtick:
$String = "video=`"Integrated Camera`""
Another way would be to escape using double "s:
$String = "video=""Integrated Camera"""
The method you chose will be down to personal preference and the readability of your code.
It's worth noting that the -ArgumentsList expects an array of strings.
Wrapping both of these up would give you something along the lines of:
$exe = "ffmpeg.exe"
$ffmpefArguments= "-f dshow i video=`"Integrated Camera`" c:\test.mp4"
&$exe $ffmpefArguments