Unfortunately, Cunning Brilliance is not very clear about what, exactly, it gives you, and what, exactly, are legal options for it. Worse still, there are abilities that are not clearly Extraordinary or not.
Factotum refers to “standard character classes” – a phrase used anywhere else in the rules, so far as I know. The general consensus is that this means any base class, but it’s certainly not the only possible interpretation. It might just mean any class that’s legal in your game. It could conceivably mean only Core classes, or even just Core base classes.
For the purposes of this answer, I’m going to go with the general consensus and stick to base classes. I’ll mention the best Core options available just because if nothing else, we can be pretty sure those are in play.
There are two questions here.
First, how does one handle “grouped” abilities? Rogue special abilities, for instance, are all listed under one class feature that Rogues get repeatedly: can you take the “special abilities of a Rogue” as a class feature, and therefore get four of them, as a Rogue 19 would? The ability explicitly says that you get the feature as would a character with levels in the class equal to your Factotum level, even above 15, provided you first get the ability at or before 15th.
Second, do abilities “default” to Extraordinary if no other qualifier (Su, Sp, etc.) is listed? There’s evidence to suggest that they do, even though this was almost certainly unintentional. In fact, in some books there are creatures who are listed as having the Extraordinary ability to cast spells – as in, cast spells the same way a Wizard or Sorcerer does, as spells, but the the ability to cast spells is Extraordinary. Since the Spellcasting class feature of various classes is never given a tag (Ex, Sp, Su, etc.), does this mean you can choose “the Spellcasting ability of a 19th-level Wizard” for Cunning Brilliance?
For the purposes of this answer, I’m going to say yes to the first but no to the second when it’s an ability that grants access to obviously not-Extraordinary abilities (i.e. you can argue that Spellcasting is an Extraordinary ability, but even if it is it’s an Ex ability that gives access to non-Ex abilities, spells, it seems to fall well outside the intent of Cunning Brilliance).
My answer will basically take this from the most liberal reading to the most conservative one.
Dubious Ex abilities
These probably aren’t legal, but some people make the argument that they are. They have additional problems beyond their legality, though.
Spellcasting is the most powerful class feature in the game, and if you can get it on a Factotum (who has the incredible Cunning Surge), that’s fairly devastating. I certainly wouldn’t allow it in my games unless the Factotum seemed to be really struggling to keep up with his teammates.
The other major problem is that you only get the ability for a minute, and you therefore do not have time to prepare spells or even to refresh spontaneous spell slots. That eliminates the Archivist, Beguiler, and Wizard, which were probably your best choices.
The Sha’ir class from Dragon Compendium, however, can prepare spells in a few rounds. They’re Charisma-based and require a lot of Diplomacy, but those are pretty good things for a Factotum to have. Thus, you could use Cunning Brilliance to get Sha’ir spellcasting, and that’s pretty awesome.
A note: the ability to manifest psionic powers is explicitly a Psi-like Ability (Ps), not Ex. Which is a shame, from this perspective, since you can refresh PP in just a full-round action after resting, and Psion powers would be awesome for a Factotum.
Not as powerful as spells, just as unlikely to fly, and also still suffering from the 1 hour preparation issue. The only reason I mention it is because they are pretty good, and you could use Shape Soulmeld to get some before using Cunning Brilliance, and use the Essentia you get for them. Doesn’t wind up being a particularly good use of Cunning Brilliance anyway. From Magic of Incarnum.
Definitely Ex, questionable “grouping”
The general consensus is that these abilities are legal for Cunning Brilliance, but I’d double-check with my DM before using any of them.
Warblades only get access to Extraordinary maneuvers, which make them pretty safe on that front. Whether you have to spend all three Cunning Brilliances on getting the Maneuvers Known, the Maneuvers Readied, and the Stances, however, is another matter – using all three like that is a much less good deal. And once again, maneuvers need to readied, and that takes five minutes: four more than you’ve got.
The solution here is to dip into an initiating class at level 9 or 18, and take the Adaptive Style feat. Now readying takes just a full-round action. I suggest that you dip Swordsage for some of the nice Supernatural maneuvers he gets, since you won’t be able to get those from Cunning Brilliance, and for Assassin’s Stance. Missing Factotum 20 is no big loss compared to what this dip can get you. It’s a good idea even if you aren’t allowed to use Cunning Brilliance to get maneuvers.
Initiators are found in Tome of Battle: Book of the Nine Swords.
Marshal Auras are explicitly Extraordinary, and explicitly introduced all within a single class feature that’s gained at level 1. This is probably one of the best answers, and one of the least dubious for those that give you so much. Ironically, though, the Minor Auras are probably better than the Major ones. This is another option that works best for a high-Charisma Factotum. Marshal is from Miniatures Handbook.
Once again, all granted by a single, Ex class feature gained at level 1. Each individual item in the list is also Ex. So it seems legal, but again YMMV.
The main attraction here is Loyal Beyond Death, which you get at Factotum 20. That’s up to 10+Cha rounds of being immune to death unless they destroy your body. That’s pretty darn good. Test of Mettle is also fairly good (one of the few aggro-drawing abilities in 3.5), and Call to Battle and/or Bond of Loyalty have their uses. Knight is from Player’s Handbook II.
Fighter Bonus Feats
If you can just choose all of those feats as a single class feature, well, that’s an awful lot of feats. Arguably allows you to take the Fighter-only feats as well, though only Weapon Supremacy is any good.
Rogue Special Abilities
A number of these are pretty nice, and you arguably get four of them. Crippling Strike, Defensive Roll, Skill Mastery, and the Bonue Feat are probably the best options.
Single-shot Ex abilities
These are almost certainly intended to be covered by Cunning Brilliance.
Rage is awesome, and Barbarians get Greater Rage early enough for you to bump yourself up to that one. Unless your DM rules that all of the __ Rage abilities form one ability, in which case you get both Tireless Rage and (at 20th) Mighty Rage, which is awesome.
(Greater) Ferocity, Whirling Frenzy
These Rage variants, from the Cityscape web enhancement “Urban Class Features” and Unearthed Arcana, respectively, are also very good.
Rogue, Scout, Ninja, and so on give precision damage in various forms: you can even stack these to get a huge amount of damage. Not really the most interesting use of the ability, but it is a powerful one. Ninja and Scout are both from Complete Adventurer.
Camouflage and Hide in Plain Sight
The Scout gets these at 8th and 14th, respectively, which is important because the Ranger doesn’t get Hide in Plain Sight until 17th. They are otherwise identical to the Ranger versions. Painful to use two Cunning Brilliances on this, but the effect is quite good.
Special mention: if prestige class abilities are allowed, the Eye of Lolth’s version of Hide in Plain Sight is ridiculously good: it’s Ex, but works like the Assassin version of Hide in Plain Sight, covering both the concealment and observation requirements of Hide. Eye of Lolth is in Drow of the Underdark.
The Scout also gets Extraordinary Blindsense, out to 30 ft. as of 20th level. Not a bad choice.
In Core only, your best options are Rage and Rogue special abilities. Fighter Bonus Feats if allowed.