Is it possible to change the value of the median based only on its value, without knowing the original set of numbers?
Example:
Consider the set $\{1, 1, 1, 5, 6\}$. The median value is $1$.
Now forget the set. I want to modify the median value by including a new element, $8$. The new median value should be $5$. Can I achieve this without the complete set of numbers?
I think not, but maybe it is possible with any auxiliary variable.
It is not possible. In fact, for any medians $m_0$ and $m_1$ and added number $x$, there exists a set with any given number of elements (larger than 1) with median $m_0$ where adding element $x$ to the set moves the median to $m_1$.
However, knowing two elements of the set closer to the median (that is, in your example, 1 and 5) would be enough to know the median after one element is added. This may qualify as the "auxiliary variable" mentioned in the question.