I love secret bosses and characters! It's that little extra touch.
Problem with secret characters is...they aren't part of the story. I for one, love character interaction. It's hugely important to every game I make, and that's one reason I can't make a game where you pick your own party (like Final Fantasy I)...I crave that interaction amongst the main characters. Maybe Tales of Symphonia's skits spoiled me, that extra character interaction. But then again, that method could and would work with secret characters, and I'd like to see that kind of effort.
One thing though, I want secret characters to stand out. Aesthetically and ability-wise. Final Fantasy 6 does this well.
Gogo is a mysterious mimic, shrouded in colorful robes, with a cloak and hood obscuring every detail. You aren't even supposed to be able to tell if it's a man or a woman! Gameplay-wise. S/he has, sub-par stats, and rather limited equipment options...But unlike every other character who has the Fight command, Gogo has Mimic! And instead of a personal ability, Magic, and Item, Gogo can use just about any and every other command available! So versatile! Takes the jack-of-all-trades thing to a whole new level. Someway, somehow, Gogo is one of my favorite characters.
Umaro is a yeti. Seriously, just a big furry berserker. He can only equip like 3 items in the whole game, and acts on his own, using only about 1-3 different attacks. But his stats are great! He's like the opposite of Gogo!
They both just stand out so much from the other characters. And that's what makes them so fun, so...cool as secrets. Also, Final Fantasy VI is one of those cool games that let you walk around as any party member! Part of the fun of these obscure secret characters is that you can even walk around as them!
Secret bosses are fun too. Secret bosses that are also secret characters are something else. C'mon, you know there's been an optional super boss that you wanted in your party. Wild ARMs did that to me.