I like Amarin much better than Mirrin with your last name - don't forget, she will rarely be using the first name, and Mirrin Miller is just too much IMO. If you want the nickname Rin, there are lots of names to choose from -
Amarina (also means rain; feels a little more feminine to me)
Rina (feels more full than just Rin)
There are really tons more! Or you could even just use Rain!
I know there are tons! But my partner is extremely picky! It's difficult getting him 'along' with names that work where we live (he has an Anime fetish! Both our boys have names from animes, thankfully those names are English).
I checked on lots of other websites and most say Amarin is Australian/Aboriginal. One said Arab.
Amarin is very close to a female name in my native country as well and I am trying to get a nod towards my own culture, but my partner doesn't generally like names from my country! So, it's very difficult. Amarin is also a bit of a "backwards" nod at my own sister, Maria (get it?). Bit cheesy, I know, but she means loads to me!
Like, I'd never get him to agree to any of those names you mention there, Lilyflower. Even though he likes Amarin, he doesn't like Amarina.
If your husband tends to favor anime/manga based named, perhaps he might like Karin (a name from the Chibi Vampire series, and occasionally pops up in other series as well). You could also spell it Carin as well.
Saorin is another anime character though Saorin is used a nickname for Saori. I think it could work.
In another anime, there's a character whose is named Isuzu but her nickname is 'Rin,' because 'suzu' translates to 'small bell' in Japanese, and 'rin' resembles the sound a small bell makes.
Isuzu (Fruits Basket :P) & Saorin doesn't really work in our native language. Marina we've talked about before, he turned it down immediately.
Perrin has already been suggested, and it reminds me of Worcestershire sauce (made by Lea & Perrin's).
I don't really like the sound of Marinell or Rinna.
Karin is a normal name where I am from and it's considered very dated.
I know a lovely baby named Taryn/Tarin, and her parents call her "Rin/Ryn" (not sure on spellings...would be personal preference). It's pretty and unique. Just a suggestion....