"It's a reference to the Bolsheviks' intent to execute Nicholas. "
I do not agree - since the house was openly called "dom osobogo naznacheniya", and it was called like that ever since the bolsheviks had Ipatiev leave it and moved in Nicholas, Alexandra and Maria, I believe it referred to the house being allocated for the imprisonment on the Romanovs - NOT their execution. Moscow must have gotten information like this within a short amount of time, and if the name refered only to the execution, then that doesn't fit with the rest of the story.
Since we all know what happened in the house, we tend to understand the phrase "house of special purpose" only in one certain way.
Like I stated, I believe that, by calling the house like that at/after arrival of the first prisoners, it only meant to state the purpose the house was used for - imprisonment of the Romanovs. Had they been put in jail, it wouldn't have been called "osobaya tyur'ma - the special jail".
Erichek