A replacement for 'Taker is definitely not a good idea, as has been said many times in this thread, it would only serve to tarnish his legacy. But introducing someone to carry on the legacy would be different, and that's something that could work. Now, I've always been a huge horror movie fan (& have been lucky enough to work on several), and I've always thought that a movie based on their back story would be awesome. Hell, I've even written up some ideas for one that I had an agent pitch directly to WWE. So maybe I'm a bit biased towards this concept. But that being said, I would definitely like to see some kind of progeny brought in to inherit the role that those two play so perfectly.
Please don't confuse that with bringing in a clone or a rehash of the character, because there is a distinct difference. Another brother wouldn't work & quite a few guys have already explained why. Even a son would be pretty hokey, and a cousin, well, no offense to anyone who suggested it, but that would be just plain ridiculous. The guys who hit on the idea of a student or a disciple, however, got the nail right on the head in terms of a plot that's both as realistic as possible given the characters, and plausible. There are a number of ways such an idea can be executed, and with an idea like this, it's the execution that will make it either successful or a failure, not the concept itself.
I feel the way I would go about it is not only plausible, but would also go back & tie up a loose end from a few years ago that left many of us scratching our heads, thinking, "WTF was the point to that?" I'm talking about the Fake Kane storyline they ran with Drew Hankinson/Luke Gallows. When it started, it was an interesting idea, as it delved back into the history of the Brothers, but it ended so abruptly, it was given minimal explanation, and was so anti-climactic that it's now looked upon as forgettable & stupid. Still, the idea of a character who was with Kane in the asylum, that even he fears, is the perfect foil for Taker to bring up to continue his reign of darkness once he decides it's time to rest in peace himself.
Have him be the one behind the fake Kane attacking the real one for the purpose of either finding or creating an even bigger monster that would be worthy of having the powers of darkness bestowed upon him. And get a different performer to take on the role as well - - since Gallows wore the Kane mask, they could get away with using anyone as long as they're younger, the right size, committed to the part, and can convincingly execute the moves and mannerisms that Kane & Taker have in common. I definitely would NOT want to see Gallows reprise the part, and don't think the character should return wearing Kane's mask either. Maybe a variation on it, coupled with ring gear similar to Taker's. Have him utterly destroy Kane, at which point, Taker can claim, "Now, you're ready".
A cool visual to symbolize the passing of the darkness & introduce this character (or re-introduce him): Taker gets buried in a casket. Lights go out. Lightening begins to strike, thunder rumbles. The Druids come out carrying torches & surround Taker's grave. The ground shakes & a fist bursts forth from the dirt. A large, dark figure hauls himself out of the ground. The audience is expecting it to be the Deadman. He raises his arms in familiar fashion, lightening hits, columns of fire erupts, the lights come up and it's the Undertaker's prodigy. Taker appears on the Titantron, imploring him to go forth, wreak havoc & spread the darkness....