Okay so I'll post what I personally think and voted for.
Strongest Skill: This was probably the hardest decision. I found that all of the skills in this game were very useful, whether through regular use or as skill checks in speech. I found Obsidian balanced the skills and skill checks in game almost perfectly for RPG games. So for strongest skill, I finally decided on Medicine, as i found basically all of my characters used medicine to some degree, and most of them i end up leveling it pretty high. So just because i found that this was so universal for all of my characters, i decided on it being the strongest skill in my opinion.
Weakest Skill: Again, very hard choice. I ended up choosing Barter, even though I think the skill is great and has many uses, I find that it seems to have some of the lowest amount of meaningful skill checks other then more money. There are some decent skill checks with it, but i find I just run into them less often than other non combat skills.
Strongest Special: This was a much easier to answer for Fallout 3, but again I find New vegas did great balancing on all of the special stats in this one. So again this was a harder choice for me, but ultimately I say Intelligence as my vote for strongest for the extra skill points at lower levels (i realize by level 50 most skills are maxed anyway, but im talking about the meat of the game, when you're lower to mid levels) and i find a lot of skill checks that require intelligence, as well as a lot of useful perks.
Weakest Special: For this one, I decided that Perception was the weakest. Whenever i was dumping a stat with new characters, i found most of them dumped either perception or charisma in favour of boosting other specials. I'd say perception is about on par with charisma for me, so it was just a flip of the coin between the two for me honestly.
Strongest Trait: I'm gonna go with built to destroy, the critical % chance is great and the negative isnt that bad, just repair weapons more often or shoot less.
Weakest Trait: Logan's Loophole. I've never seen the desire for this one. Level capped at 30 and all it does is make chems last twice as long and you cant get addicted? I mean that's nice and all, but not worth 20 levels sacrificed. I find more then enough chems to use in game and dont find it as a burden to use fixer or see a doctor every now and then (im usually going to the doctor regularly to get limbs fixed anyway cus im cheap with my doctors bags and dont like to carry too many).
Strongest Faction: I'm gonna go with House. He's got an army of robots, and can always build and fix more, much quicker then the other guys can replenish their armies.
Weakest Faction: Indy in my opinion. While it's essentially the same army as house, it's really not run by anyone except the courier (if hes even around), and more just self regulating police. There's no organization, at least with house he can connect to his entire army at any time and coordinate defenses and other such things.
Strongest level 50 perk: While ive never really made a good person and got that far, I'd say Thought You Died. Potentially 100 extra hp, 10% overall damage and immunity to criticals, great combo if you're committing to being good (which is ridiculously easy in this game).
Weakest level 50 perk: I guess im gonna go with Aint Like that now. The attack speed is definitely badass, but the ap regeneration and ap reduction on weapons never really appealed to me, i rarely use vats as it is, so i dont care too much about AP.