Distant counter?

A lot of it depends on which units you may or may not have an overabundance of. Personally I don't use armored units much, otherwise I'd go for Amelia (she's supposed to be really good with it). I gave distant counter to my Eldigan and he's an absolute beast defensively with quick riposte. Ayra might be pretty good too, but I can't vouch for it as I don't have one.

HOLY CARP what a nice stable for someone that's F2P! I've paid on MANY of those banners and came up dry every time (Amelia, Ayra, P!Azura). With that in mind, I can't claim to be an expert on your heroes, but here's my opinion on each:

tl;dr; I'd either give it to a blue armor troop or hang on to it. It's such a rare skill, there's no problem hanging on to it until you get the PERFECT hero.

- Amelia (+Atk - Hp) - She's really in the same boat as Hector, being an armored axe unit. Not a bad pick, but I'd only give it to her if you're tired of using Hector or find that she out-performs Hector.

- Ayra (+Spd - Def) - I've never used her, but is she defensive enough to take advantage of Distant Counter? If she can take a hit, she may be able to work with Vantage similar to Hector, but if she's too squishy, DC may go to waste.

- Azura P! (+Spd -Hp) - Same as Ayra, I'm afraid she'd be too squishy to take advantage of DC. That, and a dancer's job isn't really to take hits, is it?

- Chrom W! (+Def -Hp) - Same as Amelia

- Effie (+Hp - Res) - This is a pretty good pick, since there aren't many blue armored units. The added bonus is that it's an unexpected skill which can net you extra wins in Arena Defense. I've fallen to DC Effie many times because I assume she isn't carrying DC.

- Eldigan (+Spd - Res) - An OK pick, but if you've got other ranged horse superstars (READ: BowLyn & Reinhardt), you don't really need to have Eldigan in a Distant Counter situation. Bonus for Eldigan is the Weapon Refinery: you can give him DC and keep Fury with his weapon's special!
All that said, I use Eldigan on my Horse Emblem team, and I would hesitate to give it to him because of his low RES. He just melts to anything magical, which doesn't synergize with DC.

- Robin M W! (+Def -Atk) - Same as Effie, I'd pick whichever one you like more. In a tie, W!Robin gets it for having a higher BST.

- Sheena (+Def -Hp) - Same as Amelia, but lower stats overall make her a less appealing choice.

Don't let this burn a hole in your pocket, there's nothing wrong in waiting for the perfect troop to come out. If the last 6 months are any indication, IS will be powercreeping new heroes about every month or so. You never know if that 1 year anniversary special hero's kit would go perfect with DC!

Effie, Amelia, and Ayra are some of the top 10 best users of DC, so I would definitely reccomend you give it to one of those you enjoy/ use the most.

I would kill for a DC on my +Def - Hp Sheena, but I am planning to merge her with a +Def - spd one someday... With Guard and QR2 Seal, Ignis, and Slaying Axe forged for res, she becomes a high def/high res BOMB. She is the best defensive DC user in the game, in my opinion. She can kill all of the characters in your list of candidates if she is on a defensive tile, has some buffs, or is ploy/threatening the enemy's attack.

I have an Ayra with the same bane/boon as you, and I’m going to say that I think DC on Ayra is overrated and as such I’ve never given her the skill. The problem is that whenever I’ve faced an Ayra with DC whilst using Reinhardt, Rein ORKOs her with no effort. Given that Reinhardt is such a prominent opponent... she’s better as a player phase offensive unit.

That aside, I have DC on both P Azura and Eldigan and it’s particularly good on Azura who has solid resistance and can wall and kill Reinhardt and other blue mages. DC Eldigan operates similar to Xander and with the fury upgrade on his weapon, he is really formidable.

Finally, all your armoured units can use DC really well, particularly Sheena who has great resistance and W Chrom who has such a huge HP pool he’ll survive most mages first attack and kill on the first counter.

Either Sheena or Effie due to their availability for merging.

Frankly, at your current stage, unless you have a personal favorite, it really doesn't make sense to commit a DC until you decide on which hero to +10.