1, Oil (Libya has the largest reserves in Africa), Gas (23rd largest reserves in the world), Water (the largest fossil water aquifer in the world), and any other resources there (Need I say more on the control of the worlds natural resources?)
2, Wheelus Air Base. Wheelus’ location and climate made it for about two decades indispensable to Air Force operations until Gaddafi gave them the boot.(The planes are back where's the base?)
3, Arms sales to a military that now needs to be rebuilt (cha ching). There was also 4 billion in Russian contracts to sell air defense to Libya so this was preempted. (Comrade do you have a dollar the Americans stole my lunch?)
4, Control of the Med (To the Shores of Tripoli! shhh Marine Corp Hymn)
5, Pay back for allowing insurgents to go to Iraq, denouncing Imperialism, arming militants, etc etc etc. (You SOB!!! Now you're gonna get it.)
6, Africom NATOs "Anti Chinese African oil Army" has been rejected by Libya thus far. It is one of only three of the fifty four states in Africa that have not signed on. A new elite will fix that. (At the start of the war 30k Chinese industrial workers returned home. When they got there they turned on the TV to see the Americans eating their lunch.)
7, For the US it was at least partially a quid pro quo for French and British assistance in US wars. (Thanx Guys! Boom!)
8, Oil for Gold Dinars(why even mention any more)
So the US, France, and England do have interests worth pursuing while they're conducting the humanitarian kinetic military action. Unless screwing Russia and China while reaffirming western alliances and securing control of natural resources, shipping lanes, military facilities, all while making a boatload of money via weapons sales and reminding adversaries that they don't put up with any shit are things that are no longer interests of theirs.
It ain't easy being embedded together!