Nino, Eirika, or Reinhardt?

Personally, I would go with Eirika. She's a very powerful support unit that could work in a lot of teams due to her great buffs. Sure, she's a sword lord, but she can still hold her own despite being another generic red unit.

However, Reinhardt and Nino still do function extremely well as they are. Reinhardt is especially deadly on a Horse Emblem team but can work just fine on his own with his native Brave tome, while Nino is just a flat-out amazing mage to run if you use Hone Attack.

Nino will still be fairly fragile. I would wait until you have a +Spd or +Atk so she can fulfill her role as a glass cannon more effectively.
Reinhardt has a low HP pool and would prefer having a +Attk nature if possible. +Def allows him to reach 30 defense, but with 23, he'll fall pretty easily regardless of HP.
Eirika is is your best choice. Her HP might be low, but it can be remedied with an HP if so desired. Her Resistance is great for suppressing a lot of the green mage threats like Sonya, Merric, and Nino in general. Overall, she has the ability to buff all 4 stats at once with her legendary weapon and having Rodericks assist or using the Fortify Res 1 seal for the S slot.

Go Eirika. She's a keeper and she won't let you down.

Optimally, you'd want (+SPD) on a [Life & Death] or [Fury] Nino. I often see [Fury] Nino's with (+SPD/-RES) with people swearing by her because, in this form, she can actually take a hit (singular) pretty well. I'd honestly say to wait for a better unit to upgrade than those 3 you're considering, and Nino needs a buffer like Eirika to go to town... I'd say go with Nino running an "I can take a hit!" set if you really want to fill that team spot bad.

When I say wait for a better unit I meant better IV's. If your Eirika had (+SPD/-HP), if your Reinhardt had (+ATK/-SPD), or if Nino had (+SPD/-RES or -HP) I'd absolutely say you should build any of them, they're all optimal units to invest in that fulfill a huge amount of different roles each.

Nino is also one of the better units in the game to build to 4* (+10) because of how... "easy" it is to summon her. Just upgrade one of them to 5* and give the little version the upgraded tome.

A 4* (+5) Nino is equal in stats to a 5* Nino and costs 10,000 feathers to get her there, and 30,000 altogether to feed her her upgraded tome. In the long run, a 4* (+10) Nino would cost 40,000 feathers and building a 5* (+5) Nino (the equivalent of a 4* +10) would cost 100,000 feathers.

Just some food for thought.

In terms of IVs, none of them.

BUT (very big but), if you think you can get another one with good IVs one day and merge your existing one into that new one at 5 stars. Then yes, you should go for it.

As for which one, I think Nino and Rein might be good choices. Rein is for his simplistic one-shot play-style without support, Nino for her iconic Blade build which requires team support (which you have) in order to deal even more damage than Rein I believe, and finally Eirika is great for one thing, providing support and buffing Blade tomes (where her IV doesn't matter). If it was me, I would go Rein or Nino to make things like the challenges or TT much easier. They also work great in Arena despite their low BST.