Here are a few ideas, in no particular order. They are all JRPGs of various sub-genres. I wrote a bit about each game as well.
Bravely Default - JRPG
Is a bit of a throwback to some older Final Fantasy games, at least in regards to the job system. Essentially, you can change your "class", at any time. You can easily adapt to any situation, say you need another healer, well you can swap to have another healer. The story is also pretty interesting.4
Tales of the Abyss - Action/Adventure JRPG
A classic "Tales of" series JRPG, originally released on the PS2, and to be honest, one of my top 10 JRPGs. The story has many twists and a lot of depth in it, and you can really see the characters develop and grow as the game progresses. The battle system is a little outdated in regards to the "Tales of" series, but is solid.
Fire Emblem Fates- SRPG
There are plenty of Fire Emblem games on the 3DS, but this is the only one I've played, so I'll recommend that in particular. If you enjoy tactical strategy, then Fire Emblem is an obvious choice. This game actually has THREE versions. I'd suggest Birthright if you're new to tactical games, or Conquest if you enjoy a challenge. The main difference in the versions is really only story and difficulty.
The story itself is a little bit... Sparse, I suppose one could say. It's a decent story, but there is a bit more of a focus on the tactical aspect. There is also a small dating sim aspect.
Etrian Odyssey 2 Untold: The Fafnir Knight - Dungeon Crawler
Like with Fire Emblem, there are a few Etrian Odyssey games, but this is the one I played. A first person dungeon crawler, where you explore and draw the map yourself, making notes about special events, secret passages, etc. It has a story mode and a "classic" mode, where you just make your own characters. The story is decent but like Fire Emblem, there is a bit more focus on the gameplay aspect.
Shin Megami Tensei IV - Dungeon Crawler
If you're familiar with any SMT games (or the Persona series, which is a spin-off), then this is another solid entry in the series. The difficulty can be a bit unforgiving for newcomers to the series though. It also has a sequel, SMTIV: Apocalypse. VERY heavy on the story, many different contrasting themes in the series.
Shin Megami Tensei: Devil Survivor 2 Record Breaker - SRPG
Another spin-off series originally on the DS. There are only two "Devil Survivor" games. Both are good. In terms of gameplay, DS2 is more refined, with more features and polish than DS1. There is a almost total focus on the story and character interactions, including a social link type system such as in the Persona series.
Rune Factory 4 - Adventure/Simulation
JRPG version of Harvest Moon, pretty much. Fight monsters, save the world from evil. Or just farm and get married and have kids.
There is a decent story and interaction between characters.
Radiant Historia: Perfect Chronology
Whats that? Radiant Historia on 3DS? There is actually a 3DS remake announced for to release in English, early 2018. The original game on the DS was a lot of fun with a unique story, so I would tentatively suggest to pick up a copy of the remake sometime. Or even the original DS version, if you can find it.
