A random name generator (like a random number generator) applicaiton is used for this at the the DOD/National level. They are reviewed to ensure they have no connection to the reason, political, economic, geographic, cultural, or force structure to be utilized. Normally they are also known by numeric code based on regions.
At lower levels the Operations or Commander will come up with a name. Best example of 'Operation Red Dawn' when the 4th ID rolled up Uncle Saddam.
Once a CON/OP Plan is activated a more PC/PR name is normally selected.
I named an exercise -- "Razorback Guardian" -- an exercise conducted for the Arkansas National Guard (Razorback and Guardian from that) by the Readiness Group (led me to the initials "RG" that I had to find a name to fit) I was assigned to.
During the recent war with Iraq, I named three engagement areas Fist, Hammer, and Knee (all EAs were named after martial arts terms).
I've named hundreds of phase lines, assembly areas, operations, exercises, etc. Usually there is some kind of pattern. I'll demonstrate a quick example:
1. Logistics support areas might be named after truck manufacturers (Volvo, Kenworth)
2. Assembly areas after weapons (Mace, Spear)
3. Phase Lines after women's names (Sara, Misty)
4. Main Supply Routes after states (Maryland, Washington)
5. Objectives after countries (Germany, Egypt)
6. Operations after Rivers (Potomac, Hudson)
That's a quick list, just to give you an example. In accordance with the order from above, the 52nd Mech Division (most famous in the Army) launches Operation Suwanee. It goes to AA Sword, draws fuel from LSA Ford, then moves forward to PL Victoria. At H-hour, they advance toward OBJ Hungary. Their logistics tail follows the combat troops on MSR Alabama. Of course, they are victorious. The 52nd Mech is always victorious.
PS - For the unitiated, the 52nd Mechanized Division is a notional division used for all the training in the US Army. It is part of the equally notional XX Corps.