If I give my companion a unique version of their default weapon, they usually use it. Veronica uses the Paladin Toaster, which looks pretty cool. Boone will use the Bumper Sword (not unique, I know) for his melee and I'm sure would use the Gobi Campaign Scout Rifle, and Lily will use Oh Baby! if I give that to her. They'll use explosives too. It just depends on what they like.
I only used Lilly for the nightstalker quest, but I didn t have a melee weapon big enough for her to use. I gave her the assult carbine, and when it was on her back I swear to goodness it had a big giant silencer on it.
The dude is right about the strength thing. I noticed that right away. Veronica seems the most picky. She will not use a shot gun, but hook her up with a plasma rifle and she s all about it. I ve never seen boone use an energy weapon. They will all use the All American over anything. Well I dont know about Gannon.