Class System: Even though the game has developed some interesting and memorable character development systems (Materia, GF Junction, Sphere Grid, the precursor to SaGa's character development system, Crystarium), Final Fantasy was developed in a base of the tried-and-true class system. Many of the "gaiden"/spin-off games actually have class systems in one way or another (FFX-2's Dresspheres are class features by a different name; FF Dimensions and Bravely Default, as well as all FFT games, have class systems as well). Final Fantasy Tactics A2, and to an extent FF Dimensions/Bravely Default have refined the class system to allow you a wealth of options while keeping it easy. One fixed class feature or move set, one or two Reaction abilities, one or two Support abilities...and other class features granted by the class you have. I'd love to see some Synergies between classes (I.E. having levels in White Mage grants a minor spellcasting ability for a Paladin class, and same for Mageknight and Red Mage). Even games without classes have something along the lines: Cloud's a Soldier, Aeris is a Geomancer/White Mage, Squall is a warrior-type SeeD, Quistis is a Blue Mage (and so does Quina), Tifa and Zell are Blackbelts, and so forth. Even a game as FFX has classes in some way, as the Sphere Grid suggests one part while the character's Overdrives suggest another (Time Warrior/Summoner/Black Mage/Gambler/Samurai/Mechanist/Blue Mage).

Split-Party Instances: This is one part of FFVI that I've rarely, if ever, seen repeated. Final Fantasy VII was the only other game that had such, but only at the last part of the game (specifically against Bizarro Sephiroth). Dungeons where you need to split the party are interesting, since it involves careful consideration of party members in order to succeed on some challenges. It also reminds me of the Fork Tower of FFV, which you had to complete simultaneously. However, it's in FFVI where it's pretty much exploited, considering the final dungeon (Kefka's Tower) HAS to be completed with three parties.

Summons as Pokemon: It's a funny way to mention it, but FFX had some of the best examples of how Summon Monsters should behave. Somewhere between the Summons of FFX and the Guardian Forces of FFVIII (where the junction of GFs gave you class features, except Summons would get those abilities). If there's a Summoner class, allow an option for "High Summoning", where they replace the Summoner; if not, then make Summons an active part of the party. Also...

Summons as NPCs: Seeing Ramuh in FFVI be part of the story made Summons a living part of the game, instead of another reference. I mean, even how Summons work in FFIV is surprisingly interesting, even if they're only summoned to deal a super attack and that's it. Once again, FFX hit the mark on Summons, even if the only noticeable Summon-as-NPC was Bahamut acting as the phantom kid. He advanced the story, and was someone important. Given how current RPGs value character interaction, it'd be interesting to have Summons converse with the main characters every once in a while; they're as every bit a part of the group as the group itself. Speaking of which...

Character Interactions, Private Actions, Etc.: Characterization in Final Fantasy hasn't advanced that much, though I believe Final Fantasy XIII was a step towards that. However, it's nowhere near the level of characterization you'd find on a Tales of... game, or even Star Ocean. Even Western RPGs are getting the hint, with Dragon Age and Mass Effect adding interactions between characters while you're walking, speaking of topics from their own point of view. Be it from a comedic point of view, or a more serious one, those few bits (both figuratively and literally, speaking in terms of storage) liven up the story, develop it beyond mere cutscenes. Final Fantasy games have various ways to provide such moments, such as when traveling through Airship.

An Innovative Battle System that still remains Turn-Based: Active Time Battle was, at its time, an innovative combat system, but it was still turn-based. Current games have taken the game towards real-time action, and while I find it would be great for a spin-off, making it part of the numbered games is somewhat distorting the feel of the old games. Trying to make Final Fantasy's battle system a Dragon Age rip-off, just because there's a lot of people who like the Kingdom Hearts battle system, hurts more than it helps; it may cater to a wider audience, but if that audience can't be captured despite the change, you're left with a disgruntled group of fans that'll also refrain from gaming. Not to mention, given Squeenix's penchant for pretty graphics, it means a very graphically-intensive gameplay, which means further development. Turn-based games can be interesting if you add something to kick it up a notch: games with lengthy names such as Super Robot Wars Original Generation Saga Endless Frontier (and its sequel, SRWOGSagaEF Exceed) and Penny Arcade: On the Rain-Slick Precipice of Darkness Episodes 3 & 4 show how a turn-based system can be turned interesting, and at the same time intriguing and fast (through frantic combat or through tactics). Even games with smaller names like Grandia, or the first Valkyrie Profile, are pretty interesting despite being turn-based. Leave real-time to KH, or maybe even the Mana series.

Your Decisions influence the Story: A tangent to Story, but let's look at it at the concept of Mechanics. Perhaps a class system could allow for some sort of class tree that depends on your decisions in story. Perhaps you hold in your fate the destiny of two cities, and the choices you make will save one, save both, or save none. You can see this, to an extent, in FFVII; dealing with the Huge Materia meant the difference between a friendly, safe Corel or a ruined, bitter one; the difference between watching how METEOR can't be stopped despite sending a WMD or getting the games best Materia. Part of those choices are story-based, but some directly influence your mechanics, including the way to make your characters stronger. That degree of interaction between story and mechanics makes it interesting.

I don't speak much about Music, since aside from FFXIII (but that's not because I don't like the game; it's that I consider Hamauzu lacking compared to my Big 5), the music is usually great. However, I'd agree with Fluffy and I'd let Yoko Shimomura be the next FF composer. After all, since it's mechanically similar to Kingdom Hearts, why not have the composer of Kingdom Hearts? You could also have Kenji Ito, a great composer who's currently doing pretty much nothing (thanks to Squeenix leaving the SaGa series pretty much dead, even when Akitoshi Kawazu still wants to make games) and who's a Uematsu alumni. It doesn't have to be similar to Uematsu's work, but someone a bit more eclectic (part of why I feel Hamauzu doesn't work for FF is because he tends to incline a lot to Classical music, focusing on the harmony and only a bit on the feeling).