Ahh yes, Morrowind-style alchemy, not quite the application I imagined. Well, this too can already be done... it's just a bit more complicated.
Detect when the alchemy menu is open, edit the GMSTs to allow all effects to be used, then edit the OBME effect override data on any ingredients in the player's inventory to be named "Unknown Effect". You'll also have to keep track of the potion as it's being made, giving the "Unknown Effect" name to any of its effects that aren't known on at least two component ingredients. Not sure if you can get that done before it's displayed, but if not I believe MenuQue allows a forced in-place refresh. When a potion is created, reset its effect overrides; when the alchemy menu is closed, reset the ingredient overrides.
Of course I'd much rather see a hard-code solution myself. Just sayin', there IS a way.

Fact is, I've already got half of it written myself; I'm using contextual effect edits for several purposes during alchemy, spell creation and enchantment. But this one is particularly more complex than most, and to be honest I don't think it's a good design. All the Morrowind system does is encourage external reference, or worse, just making the potion to see what the unknown is, then reloading. (Even if you keep the effect "unknown" on the final potion, you can go one more step and drink it to find out before reloading.) Oblivion's method has the obvious flaw that a high-level alchemist might get side effects from a pair of ingredients, when a low-level alchemist can get the positive effect but not the negative one; but they were moving in the right direction IMO. Building on what we've got, I'd go for one of the following rather than returning to Morrowind style:
1. The vanilla Oblivion system, with the ingredients carefully re-designed so that there is never a situation where an Expert produces a more useful potion than a Master, etc. This is a fair bit work and has inherent compatibility issues with new ingredients; but it's not a bad solution overall.
2. My own solution, make all effects available all the time. This requires some additional balance elsewhere to keep the mastery ranks meaningful; I've changed the number of potions you can drink at once to (novice) 1, 1, 2, 3, 5 (master), and you actually see ZERO effects at Novice rank, meaning that you can't create potions: only training and Wortcraft can advance you to Apprentice. Overall I'm pretty happy with this; it's fully consistent and there are no compatibility worries. However, you could conceivably go one better...
3. Allow a high-ranked alchemist to control their mastery rank during potion creation. You could actually do this in either direction... working from the vanilla system, a Master could create potions as if he were an Expert (or lower), shaving off that nasty detrimental effect in slot 4. Or working from my system, it would go in reverse: an Apprentice is stuck with all four effects, a Journeyman can leave off the top one, etc. (This is nicely consistent with a Master's ability to use a single ingredient to create a potion with its slot 1 ability.)
JRoush, if #3 seems like a fairly simple thing to implement at your end, consider that a request; but don't let it delay anything else. I'm pretty sure this would actually be easier to implement in scripts than the Morrowind-style method... you'd definitely need MenuQue, but there's a lot less to keep track of.