There are multiple ways to do this. The easiest would be simply to put Attack or Spell Power on weapons, which would still leave them segregated by role to some extent (melee would want the AP weapons, ranged casters would want the SP weapons, and hunters would still be the only ones using bows, crossbows and guns). A more complicated but perhaps more compelling system would be to have them switch between AP and Spell Power, so that a holy paladin could use a weapon for healing, then switch over and use said weapon for tanking. Still more complex but perhaps even more interesting would be to have weapons retain int, agi or strength and switch depending on which class was using them as well as each role. If you visit the original post at 7th Tower, he breaks down how it might look using Siege of Orgrimmar as a template. The current breakdown of agility and strength weapons would, in either scheme, now be available to a broader range of players.
It would certainly fit within the paradigm of broader usefulness for gear established by the changes to armor. The question becomes what are the up and downsides to this?
The first downside that comes to mind is simple - more competition. We're already going to see a lot more competition between classes who share an armor type, as well as class specializations (no more tanking plate or DPS plate or even healing plate, no more spell leather, no more resto shammy mail) and if we changed weapons so that they switched between Agi and Str (never mind also switching between Int, Agi and Str depending on who was using them and what their spec was) then pretty much every class that can possibly use a weapon would be rolling on every weapon that it could use.
It could also bring back some of the weird edge cases. There was a time in Molten Core that most healing paladins and druids used a mace, Finkle's Lava Dredger. This kind of itemization has been brought back from time to time, with mixed results. If all weapons switched between Agi, Int and Str depending on class and spec (or between AP and Spell Power, if we went that route) then you'd see a lot of debate on this issue, with holy paladins using 2h swords and resto druids/mistweaver monks using polearms. That's not necessarily a bad thing, but it would definitely create friction - imagine your melee DPS bemoaning seing a holy paladin swinging Reclaimed Gorehowl off of Grom.
On the other hand, maybe that's a positive - no more raids bemoaning that they get too many Gurthalak's or disenchanting that sixth Invocation of the Dawn that dropped that no one wants. There would need to be some standardization between caster and melee weapon DPS, of course - right now caster weapons tend to have much lower DPS, that would have to change as well for this to work. But as much as I can see downsides, I do think a scheme like this would finally make it feasible for warriors to use fist weapons and get more variety out there for rogues and shamans, both of who are often stuck using the same two or three drops (monks end up with a similar problem).
So we turn to you, gentle reader. Good idea or bad idea?