This app was mentioned in 26 comments, with an average of 1.46 upvotes
You can play 3 levels of this Layton spin off game for free https://play.google.com/store/apps/details?id=com.Level_5.MysteryRoomENG it's relaxing and pretty fun. Not as good as the originals but still good
I'm not sure how well this counts, but I really enjoyed Layton Brothers: Mystery Room. It's actually part of the Prof. Layton series, but you don't need to have played it to understand what's going on here. (It actually barely references the old games.) Gameplay's actually a little more Phoenix Wrighty without the courtrooms, compared to typical Layton play. It was fun solving the various cases, especially the final one.
These probably aren't it, but since you mentioned Professor Layton I'll link to them anyway:
Both of those are official Layton titles on Android and iOS.
Can you suggest a game similar to Layton Brothers Mystery Room? Something in that sort of detective genre, without turning into too much of a "hidden object" game.
There is a Layton game for Android
https://play.google.com/store/apps/details?id=com.Level_5.MysteryRoomENG
I found this too: "another case solved". Looks like Layton. Not familiar with it tho
https://play.google.com/store/apps/details?id=com.noodlecake.anothercasesolved
https://play.google.com/store/apps/details?id=com.Level_5.MysteryRoomENG&hl=de
This may be something you would like I guess. (If you don't already know about that game).
I'm not sure if these require always-on internet. I don't think they do, but I'm not sure how to confirm for sure.
There's a couple of Professor Layton games available for Android if that's of any interest to you.
One game I really loved was Devil's Attorney. It's a turn based strategy game where you're an attorney using evidence and abilities to whittle down the opposition. It's a lot of fun and some of the challenges are actually quite tricky.
There's one game I always suggest to people, it's a deck-building card game called Ascension. You can play for free, or if you want to you can buy card pack expansions. You're not at a disadvantage by not paying though, as you play with a central deck of cards. I'm not sure how familiar you are with deck building card games. The Android app for this one is really solid though.
As a puzzle game, I also highly highly recommend Lara Croft GO.
Arrrrgh I STILL GRUDGE against 999 because as much as I loved the puzzles the climax of the story just fell completely flat for certain spoilerish reasons that pissed me off, namely that [I actually have prosopagnosia and it doesn't work like that arrrrghhsdlgkjsdlgjasdf it was all wrong, I will probably never get over how stupidly wrong that was, but anyway](/spoiler) - that said, puzzle fun plot heavy games are actually the main thing I go for, and I'm pretty sure I can help with this! Also if you liked 999, you will probably also like Virtue's Last Reward - it is a good game, all in all, and the puzzles are more of the same sort of fun.
I'm surprised you didn't mention Ghost Trick! This game is great! It's made by Takumi Shuu, who was the main guy behind the Ace Attorney series, and if you liked that you should get a lot out of this game. I would strongly recommend it if you haven't already played it!
Other great, plot heavy games for the 3DS include The World Ends With You, which has an excellent story, pretty interesting gameplay, and is a real blast. I would suggest this over Zelda, honestly. Like, I'd actually not rec that you get a Zelda game right now - they're more action-y than anything else, and while the plots are strong, the gameplay if you're not into that sort of thing can end up feeling a little frustrating. What you could do is maybe try catching some Lets Plays of it to see if it's your thing, or if it's something that you'd like to try, and make a decision based on that.
Also if you don't mind, I'd like to suggest something not for the 3DS but available for mobile platforms (i.e. if you have an iPhone or Android based smartphone, you're good!) - Layton Brothers: Mystery Room. You can find it here and here for IOS and Android respectively. It's not actually linked to the Professor Layton series, apart from the name, and the gameplay is actually much closer to Ace Attorney type investigation and murder mystery solving. I loved it - the plot is solid, the characters are charming, and the gameplay is actually very engaging. This is probably my #1 rec!