The weight mechanic, specifically the formula used in the GBA games. It uses Constitution as a determining factor, which really can't be changed unless you were to give someone a consumable like a Body Ring. Because of that, it prevents characters from wielding a typically stronger weapon without suffering from a Speed penalty.
For example, Eirika has a Con stat of 5, which means that swords with a higher weight than 5 she suffers a Speed penalty from, thereby locking her into weaker weapons like Slim Sword and Iron Sword, either until she promotes (which is much later into the game) or by giving her the aforementioned Body Ring. Sure, she's got Speed to spare due to how easily she caps out her Speed with her Speed growth rate, but I'd much rather not have to worry about receiving a Speed penalty in the first place.
I'd make it so that, like in Path of Radiance and Radiant Dawn, the formula is Speed - (Weight - Strength) where if (Weight - Strength) is negative, the result is taken to be 0, instead of GBA FE's Speed - (Weight - Constitution) formula. It makes much more sense to have a unit be able to carry a heavy weapon using their own strength instead of their literal body size. That, and that you can actually receive stat growths in Strength.
I have no problems with Constitution as a stat, since it helps determine if a unit is able to Rescue another unit or not, which I think is an already well-implemented mechanic.
I want to talk about this a bit since it's something I've been thinking about too. (I don't think there's too much of a problem with commenting on other answers?)
Personally I think it's completely valid that weapon weight should continue to affect you and your choice of weapons throughout the whole game. After all, the whole point of it is to add more depth to the decisions you make when choosing weapons. If everyone can completely cancel it out after a certain point (as tends to happen in FE9/10/11 which use Str to offset weight), then the mechanic is pointless.
The other problem with the approach of using Str is that magical classes end up disproportionately more affected by weapon weight, due to inherently having low Str growths. It's fine that certain individual units might be more or less affected by weapon weight, but it should be specific to the unit, rather than being inherent to their weapon type.
With Eirika, for example, her low Con means you have to account for that when picking weapons for her. Do you go for the lighter weapon that doesn't deal as much damage, but will let her take full advantage of her Spd, or the stronger but heavier weapon, that hits much harder but slows her down much more? Both approaches can be valid depending on the situation, so it makes you think more about the enemies before you, and which approach works best. It's also another way of balancing her high Spd, outside of simply throwing faster enemies at you, so it gives the designers a little more freedom when designing maps and challenges. If you can simply ignore weapon weight entirely, then all of these design considerations are invalidated.
So I think Con being static throughout the game isn't such a bad thing, from a game design perspective. Though I still think it would make a certain amount of sense for a unit's capacity to wield heavier weapons to increase over time. Thracia 776 had a growth rate for Con, which would seem to solve that, though I'd personally have Con growths average out a little lower than they did there (like, no higher than 10% or so). You could also play around with having more ways to increase it. Maybe using weapons that exceed your Con enough times can get you a 1-point increase in Con, or something.
What's your favorite chapter/map in the FE series, from a strategic standpoint?
Oh hey, we've got
a whole thread for that one already. :p
I'll again say Elincia's Gambit, the Endgame of Part 2 of Radiant Dawn. It puts you in a weak position (notwithstanding the fact that you have Haar), and tasks you with either walling up and holding your position, or rushing for the boss and offing him quickly, and like with a lot of such maps the latter is often the best way to handle it. It also makes some of the best use in the game of mechanics like height advantage. In general it's just the right amount of challenge, without getting too frustrating; if you fail there's usually some other way to approach it that'll let you do better next time.
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I'll ask the opposite of that one. :p
What's your least favourite chapter, and what would you do to improve it?