First off lets look at the general gameplay of the Layton series and how Sakurai tends to design most characters.
F-Zero for example, is a game all about high speed, high risk and high reward, this is reflected onto Captain Falcon's moveset. While he doesn't physically fight himself. His moveset does revolve around getting close to your opponents and dealing lots and heavy damage as fast as possible but with very tight windows. He is also one of the fastest runners in the game (behind Sonic of course) which also reflects his series high speed nature.
Animal Crossing is all about collecting as Sakurai stated back when Villager was first revealed. Thus his moveset reflects upon the various tools they can collect and use in their game. But most notably their neutral special. Which allows them to pocket all kinds of items and projectiles that come their way.
Duck Hunt was a game all about the Zapper peripheral that came along with the game, and being able to shoot various objects straight at the screen. Their moveset is reflected upon that, along with two other games that where compatible with the Nintendo Zapper.
These are just a few examples, for more in depth analysis on these characters plus more and how their designed, I think the best place to look for that would be the Know your Moves series done by RelaxAlax over at his youtube channel.
But back to Layton.
The Professor Layton series main core mechanics are the puzzles and brain teasers. Their often designed in such a way to make the players think close and hard about what the questions or tasks given is actually asking for. For example, the infamous mouse puzzle from Curious Village gives you a general overview of how mice breed but then tells you that you bought one mouse and then proceeds to ask you how many you would end up having after a certain amount of time has passed. The answer would be one as at the end of the day, you only bought one mouse. These sort of puzzles that are designed to make the player make quick assumptions without reading the question given carefully is a prevalent part of the Layton series.
So how would this reflect upon Layton's moveset? It should mostly revolve around mind games and making your opponent have to think ahead in terms of what you would do next. Making quick assumptions of what the Prof would do next will mostly lead to getting thrown off into a disadvantage.
So back to Layton's moveset, here are a few examples of attacks the Prof can use.
Side Smash
These two puzzles "The winning Arrow" and "One true Sword" are about you having to figure out the one real object among the number of fake ones.
One move Layton could have (Possibly most suitable for his side Smash attack) is him shooting three arrows at his opponent. However, only one of them is real and cause damage, the other two are fake and will just disappear with a poof with no damage or knockback given.
You are able to choose which of these three arrows are the real one. This would be a particularly handy tool to use for edge guarding and syking out your opponents who are trying to recover back to the stage.
Side Special
This puzzle involved you having to roll a sleep bomb through a panel maze in order to reach an open vent. One of Layton's moves can involve him shooting out a sleep bomb that can put your opponents to sleep. You will be able to navigate the bomb at 90 degree angles about 5 or 6 times each (Much like Snake's side special or Ness's up special) This way you can change its course quickly and have it hit your opponent from above, under, or the back instead of the front. This would most likely be useful against counter/reflect users, Most particularly characters like
that absorb or reflect projectiles at a close window.
This are the main examples of puzzles Layton can use that reflect closely around the gameplay of his series. I'm sure there are more throughout the series history that Sakurai can use that expand further around this idea.
Other puzzles they can use are as follows.
Tether
The Grand Hall Rescue puzzle could be used as a tether recovery. Layton will swing with a candle and rope towards the ledge.
throws
His throws could have him solve a quick slide puzzle, as the opponent would then be thrown at whatever direction is open once the puzzle is complete. The picture could be the opponents Stock Icon, just like Mr. Game & Watch's throws.
Forward aerial
Back aerial
Layton will launch off the fish from Last Specter. He will be able to control its angle too. It will bounce off any walls it hits
Down air
Layton would throw down the toy car from Lost Future, and will then move forward after landing. Hitting any opponents who get in the way, you can even stand on top of it momentarily. (Basically a mix between Pac Man's Fire Hydrant and the Pokeball Pokemon gogoat.)
Dash Attack
Slippery Trip is a puzzle where you have to navigate through a maze with Banana peels spread all over the place. If you walk over one, you will slip quickly towards the wall. Two ways this can be incorporated is through a dash attack where Layton will momentarily slide over a banana peel for a short bit. Might be the longest dash attack in the game too. Otherwise, this can also be a effect exclusive to Layton whenever he walks over the actual banana peel item itself.
Jab
Rapid Jab
The Magnifying glass can be used as a Jab and Rapid Jab attack. Once the rapid jab finished a big red exclamation mark will show up to launch the opponent away. This is in reference to many players tapping around on the screen when looking for hidden hint coins and puzzles especially in the DS games.
Side tilt
Layton would throw an apple as the Hamster will then run after it. If any character picks up the apple or is so much as near it, the Hamster will proceed to hit the opponent with a barrage of attacks. Much like the Sable Prince Assist Trophy.
Down Tilt
The toy robot from Miracle Mask will briefly spring up. May potentially trip up the opponent too.
Up tilt
! Up Smash
His up tilt can be him finding a hint coin, whereas his up smash will spring up a red exclamation mark.
Down Smash
Layton will use the Squirrel from Azran Legacy to roll a giant nut down across the stage. The longer you hold on the attack, the faster it will go.
Layton's hand made gadgets
Down Special
The Town Catapult from Last Specter. It might be downscaled a little. But I can see it working a bit like Zelda's Phantom, to where it gets build up over time and then shoots a barrage of Rocks directly towards the direction you've set it up. Perhaps you could have Luke standing next to it and shooting it out once its ready.
Neutral Special
The Slot Machine gun from Unwound/Lost Future, can be charged up and then shoot a barrage of hint coins towards your opponent. Or it could work like Ivysaur's Neutral special but sideways.
Up Special
The handmade Flying Contraption from Eternal Diva could work very well as an up special move for Layton. If for any reason he can't have any content from the movie. Then the Hang Gliders from Curious Village and Azran Legacy can work too.
Final Smash
His Final Smash would obviously be his famous "The Culprit is you!" finger pointing scene. Its an iconic staple of the Layton series after all. For some added fanservice and hype, they can include other Layton characters to point towards the opponent alongside him such as Luke, Flora, Emmy, Katrielle, maybe even Alfendi.
This is only the tip of the iceberg and I'm sure that there is a lot more Sakurai & co can pull from for the Prof. Other examples are the Fish minigame, Toy Robot and the Squirrel. I'll update this as I find more new idea's they could use.
In terms of Layton using a weapon like a Sword or Lead Pipe. While it is an iconic part of the Prof and some of the most hype moments in his series is him going toe to toe with a fencing match with his opponents throughout the series. I feel that making him use a Sword would be a bit of a disservice towards the character and the series that he hails from as it only represents a small part of the franchise. At the very least if he does use one, it should be used very sparingly.
An alternative way this can be used as a compromise is giving Layton a Lead Pipe as an equippable item that he can use. You know how
is able to choose which of the three starters he uses first at the Character Select Screen? Layton could have a Lead Pipe item next to his character portrait and you will be able to toggle the item on or off if you so wish. This would be useful for those who don't wish to play with items on, but still want Layton to use a battering weapon in battle
The way the Lead Pipe would work is essentially like how you would use any other item such as the Beam Sword or Lip's Stick. However, Layton would have the ability to use a few more moves whenever he holds any battering item rather than just for jabs, Smash attacks, forward Jabs. But also as an additional aerials, up smash and down smash. Layton would be able to throw the Lead Pipe away like any other item, if he loses it, it will respawn somewhere near him for him to pick up. Like with K.Rool's crown.
I think the reason of opting for Lead Pipe to be an equippable item is that Hershel himself, doesn't own an Sword or Lead Pipe of his own, but rather he picks up whatever battering item he finds lying around and uses it to his advantage. So making it an item feels like the most appropriate thing to do as that's essentially how items work in smash too.