I've done some research and experimented in the game with different companions and have come to the following conclusions. Seeing as how my methods of playing this game differ from probably everyone else, I do things differently.
Light Touch Builds: ED-E and Cass. When fully upgraded, ED-E raises your DT a little. Drinking whiskey or wasteland tequila raises your DT, depending on your survival skill level. Both these increases make up for your lower DT because you wear light armor.
Non Light Touch Criticals Builds (Long Rage): Rex and either Veronica or Lily. You need them for melee support. If you destroy the brotherhood of steel bunker in the Mr. House, Legion, or NCR routes (unless you can form a truce between NCR and the brotherhood, you may need to replay Veronica with Lily seeing as how she isn't involved with any faction.
Non Light Touch Laser Builds: Use ED-E due to one rank of his ultimate perk raising beam damage. Your humanoid companion varies depending on certain factors. If you take Meltdown, get Raul due to his ability to decrease the decay of your weapons if you are using more powerful ammo. If using rapid fire weapons, Raul is also good for the same reasons due to him staying out of harm's path. If not using either rapid fire or Meltdown, take either Veronica or Lily.
Rapid Fire: Since these weapons tend to decay at a faster route, ED-E and Raul are the best choice. One rank of of ED-E's perk allows you to repair your weapon once a day. Raul's maintenance perks also slow down the decay rate. This combination allows you to save carry weight by not carrying a ton of weapon repair kits, as long as you are in the main game of course.
Stealth: Lily's stealth girl raises your sneak attack damage criticals, making her the best choice. Increased duration of stealth boys is a welcome extra layer of frosting on the cake. ED-E's enhanced sensors can detect enemies further away, allowing you to dispatch them at greater distances.
Explosives: None or either ED-E and Boone or Raul. You don't want companions who prefer to attack in melee range.
Melee: ED-E is great for long range support. For your humanoid companion, it all depends on whom you ally with endgame. If siding with NCR, take Boone. If siding with Yes Man, Mr. House, or the Legion, use Raul and give him a powerful rifle such as Medicine Stick.
Short Range: The flamer has a shorter range than other weapons that attack from range. You need ranged support from ED-E and another long range companion. Raul is best for his ability to slow decay rate. ED-E can also repair your weapon once a day.
Arcade is the only companion you can't take with you endgame after completing his companion quest, so I didn't list him for any endgame builds.
Also, is there any point to recruiting certain companions even if you don't intend to use them endgame? For example, is there any point to recruiting Veronica if you don't need melee support or an easy entry to the Brotherhood? Is there any point in enlisting Boone or even interacting with him at all to do his first quest "One For My Baby" if you don't intend to use him end-game? And if Arcade doesn't accompany you end-game, what point is there to even getting him?