Due to what I've said about LO not increasing potential I'd like to let it be known that I don't think making it character specific is necessary. Since it doesn't actually improve the character it should not be used to help balance them.
Again, I have to agree with Matt#s. It shouldn't be used for character balance.
Although, I think there are two good reasons not to make it the same for all characters.
1. Making it character specific adds, admittedly to a small degree, an element of knowing your character. You have to know how much leeway you have in terms of timing based on your character. I think that should be preserved.
2. Some people think this is a small issue, but I think matching it to the animations is an important game design element. Visual cues are extremely important in game design. Brawl's attempt at matching the animations failed, because it leaves ledges open immediately upon crossing the plane of the stage, which can make it look like you grab while the other person is still on it. I'd say about 10 frames AFTER the animation looks like it's off the ledge would be about right for the ledge to look grabbable. Yes. I said grabbable.
I mean, in terms of good game design, it'd just be better to visually see that the ledge is open than to judge it by whether or not the other person's animation has ended. This is probably more important for the person doing the edgehogging, because you don't want them to have the reaction "what! I was clearly still on the ledge! You should have died!" I think that would be what makes a melee player unsatisfied, not whether it's melee exact.