The specific heat of mineral oil (1.67 kJ/kg.K) [1] isn't much better than air (1.0 kJ/kg.K) [2] when considering the difficulty of using a liquid. If you are going to use a liquid - water is a much better heat transfer medium (3.93 kJ/kg.K) [3].
The difference in specific heat is only a factor of 2. The big difference is in the density, which is about a factor of 1000 larger for liquids over air at atmospheric pressure.
(However, mineral oil has much higher viscosity than water, which I'm not sure is what you want as it suppresses convection. I guess it's a question of whether you take that over the non-conductivity.)
Although I can't watch the video at work - I assume it's one of the many videos where the entire motherboard is submerged in mineral oil. Certainly that's the case, and it's why overclockers have been trying this for over a decade. It's just a problem with moving the heat away from a very concentrated area of the die. As lutorm observed, the viscosity (combined with the inability to transfer heat quickly) ends up preventing an efficient mineral oil based cooling solution.
1: http://www.engineeringtoolbox.com/specific-heat-fluids-d_151... (search for "mineral")
2: http://www.engineeringtoolbox.com/air-specific-heat-capacity...
3: http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/spht.html