Lily: Lily is a creature with ST=9, EN=8, and her Health and Melee combat effectiveness are very high. (Her stealth perk is great, depending on your play style it can be a huge plus. You can get, "Oh, Baby" by doing her personal quest, it's one of the best 2 handed melee weapons in the game.)

Veronica: Tag=Energy Weapons, Science, Unarmed. ST=7, EN=6, Base Health=190. (Use unarmed power fist type weapons, obviously. Veronica has access to factioned Brotherhood T-51b Power Armor right after you hire her, just visit Repconn HQ and search the upper floor -- this makes her far and away the best early choice.)

Boone: Tag=Guns, Melee Weapons, Sneak. ST=7, EN=5. Base Health=190. (Sneak is a major factor for some players, this might make Boone better than Veronica, overall.)

Cass: Tag=Guns, Melee Weapons, Sneak. ST=7, EN=4, Base Health=175. (Can get +50 Health with the Calm Heart perk if you finish her quest. With her sneak skill this puts her more on par with Boone and Veronica.)

Raul: Tag=Guns, Repair, Science. ST=6, EN=4, Base Health=200. (No melee skill tagged, otherwise he'd be up near the top.)

Arcade: Tag=Energy Weapons, Melee Weapons, Medicine. ST=6, EN=4, Base Health=135. (LOW base health, that's why he's below Raul.)

Much depends on the weapon and especially the armor you give them, since in melee they'll take a lot more hits than they would when fighting ranged. You can't give Lily or REx armor, nor can you give Lily certain weapons (and Rex none) but Lily is still the most effective melee fighter of any companion, she has a fair shot at soloing a Deathclaw in melee.

If you consider Rex a choice (6 ST, 8 EN, rather crappy health) he tends to charge off into combat and can easily get a long distance away in a short time. This gets him into trouble so probably the best choice for his companion quest is the Legion Dog's brain because it makes Rex tougher to kill, it adds DT and health. (Lily charges off too, but she is a lot tougher to kill than any other companion.)