I honestly think it depends on what niche you are trying to fill within the team, and what sort of comp you have. I'll give you my personal views on the pro's and con's of each.
Thresh.
Pro's: Thresh can be a brilliant play-maker providing you, a) have your team on your side and following up. b) have an aggressive adc who can go hard early. and c) can land those clutch hooks onto the enemy team's carry. His mid-game damage is very high for a support, due to his soul mechanics and the ap/auto attack damage it gives him (the latter from his Flay's passive(e)). The Box (r) is arguably one of the best support ultimates for an engage/all in comp and can also function as excellent peel for any carry getting attacked. Lastly, the Dark Passage (w) is one of the most clutch gank tools/save my trapped teammate tools in the game. Throw that lantern behind you and BOOM, suddenly the jungler is on top of the enemy adc from the tri bush.
Con's: No sustain at all, for both you and your adc. So against a poke team that is nimble and can avoid your hooks, you're going to have some trouble. Also, his cool downs are fairly long in comparison to some of the other supports, so if you wiff that hook or misuse the lantern (w) then you're going to have to play passive for at least 10 seconds if not more, which can be a problem if the other team suddenly goes in. Also, especially in lower levels of play, if your team doesn't follow up you're done for. He's not like blitz where you take their carry and insert him into the middle of your team. Thresh is like Leona where you go all in, and if your team isnt with you you're done for. He has no major escape other then a flash, or a hook to a neutral monster (if you haven't just hooked to engage). Also, he is frequently banned in draft/ranked at mid/low elo play. I hear he isnt quite as perma-banned at the upper divisions.
Nami:
Pro's: Fairly well rounded and aggressive, but with the ability to play passive too. Her stun is surprisingly long and her heal is very useful if you use the bouncing mechanic well. Her attack steroid (e) can be a brilliant boost in any phase of the game, for both harass or for an all in. It also slows, so if you're trying to peel for your adc use it on yourself and then auto the enemy to slow them. Her ultimate is godly, having a MASSIVE range with a knockup, a slow, decent enough damage, and is comparable to the power of sona's creshendo, especially in the jungle. Her passive is secretly super useful also, even if it doesn't always feel like it. Spam spells on yourself or an ally to get them back into lane faster, or to run away/chase faster.
Con's: The bubble/stun (q) animation is iffy where no matter how far you are casting it, it always has the same time in the air. This can be difficult for players unfamiliar to nami to get down, but once you get used to it, it doesn't seem as problematic. Nevertheless, the bubble animation is also rather slow and a good vayne,caitlyn,ez,ect. can dodge it (so can someone with tier two boots if they're smart). Her early heal on a single target is slightly lackluster for its mana cost and cooldown (rank one and two is where this is a real problem), so you have to watch your early mana consumption. Lastly, she is squishy. Very squishy. Especially if you are running utility masteries/runes to deal with her mana problems early.
I hope some of this helps in any way. Good luck with whoever you choose.