The idea I have is somewhat similar to the multi-player part of Portal2, but completely made for single-player.
The way I see it is somewhat similar to the Broken Sword - switching main characters every now and then to unravel the story.
However, it would be nice to make the puzzles more complex so the player has to choose characters within the same area and make them work together to solve the puzzle:
Imagine a haunted house where 3 characters enter at the same time, but get separated.
You will have to control each of them individually to activate hidden mechanisms and reveal traps to help the group survive to the exit of the perilous house.
Also, each character would have their own inventory to create more puzzle options. Or even some special abilities (i.e. read some ancient writings, technical knowledge to work mechanisms and such, forensics knowledge, etc.), which would be something like The Lost Vikings.
What do you think?