Consider any cooperative TU game $[N,\nu]$ in characteristic function form, where $N$ is a set of Players $i\in N$ and $\nu:2^N\to\mathbb{R}$ is a characteristic function assigning some real worth to every coalition $L\subseteq N$. Then, let's define some concepts (all definitions are from Maschler, Solan and Zamir, Chapter 20)
EXCESS
For every vector $\mathbf{x}\in\mathbb{R}^N$ and every coalition $L\subseteq N$. \begin{gather*} e(L,x)=\nu(L)-\sum_{i\in L}x_i \end{gather*} is called the excess of coalition $L$ at $\mathbf{x}$.
Given a vector $\mathbf{x}\in\mathbb{R}^N$, we compute the excess of all the coalitions at $\mathbf{x}$, and we write them in decreasing order from left to right, $\theta(x)=(e(L_1,\mathbf{x}),e(L_2,\mathbf{x}),...,e(L_{2^N},\mathbf{x}))$, where $\{L_1,L_2,...,L_{2^N}\}$ are all the coalitions, indexed such that $e(L_1,\mathbf{x})\geqslant e(L_2,\mathbf{x})\geqslant ...\geqslant e(L_{2^N},\mathbf{x})$.
SET OF PREIMPUTATIONS
Given any game $[N,\nu]$ the set of vectors $\mathbf{x}\in\mathbb{R}^N$ that satisfy $\sum_{i\in N}x_i=\nu(N)$ is the set of preimputations of $[N,\nu]$, denoted $X^0(N,\nu)$.
LEXICOGRAPHIC RELATION
Let $a=(a_1,a_2,...,a_d)$ and $b=(b_1,b_2,...,b_d)$ be two vectors in $R^d$. Then, $a \geqslant_L b$ if either $a=b$, or there exists an integer $k$, $1\leqslant k\leqslant d$,such that $a_k>b_k$, and $a_i=b_i$ for every $1\leqslant i<k$. This order relation is termed the lexicographic order.
PRENUCLEOLUS
The Prenucleolus of the game $[N,\nu]$, denoted $\mathcal{P}(N,\nu,X^0(N,\nu))$ is the set \begin{gather*} \mathcal{P}(N,\nu,X^0(N,\nu))= \{x\in X^0(N,\nu): \theta(x) \leqslant_{L} \theta(y), \forall y\in X^0(N,\nu)\} \end{gather*}
For more details about these concepts, see Maschler, Solan and Zamir, Chapter 20. Now, take any coalition $L^{*}\subsetneq N$ of some game $[N,\nu]$ and define a new game $[N,\nu^{\prime}]$ in which $\nu^{\prime}(L)=\nu(L)$ for all $L\subseteq N\backslash L^{*}$ and $\nu^{\prime}(L)=\nu(L)+\epsilon$ for some $\epsilon>0$. Then, my question is the following:
Does it necessarily exist at least one Player $i\in L^{*}$ for whom $\mathcal{P}_i(N,\nu^{\prime},X^0(N,\nu^{\prime}))>\mathcal{P}_i(N,\nu,X^0(N,\nu))$?
My intuition says yes, but I can't prove it myself. If the answer is negative, a counter-example will be enough. If it is affirmative, an intuition of why that is the case will also be enough.
Your intuition is not in general true. Due to the work of
H. Reijnierse and J. Potters. The B-Nucleolus of TU-Games. Games and Economic Behaviour, 24:77–96, 1998,
it is known that for zero-normalized games with v(N) > 0, a collection of at most $2 (n - 1)$ coalitions admits the determination of the (pre-)nucleolus. Thus, changing the worth of some of these coalitions will change the outcome of the (pre-)nucleolus. However, those coalitions which do not determine the outcome of the (pre-)nucleolus can be changed without harm. Or to put it differently, at most $2 (n-1)$ coalitions are needed to replicate the (pre-)nucleolus, the others can be discarded.
Of course, the hardest work is to determine these coalition in advance. This is ongoing research.
More details to this topic can be found in an extract of my new book project that can be downloaded from the following URL:
Chapter: Replication of the Pre-Nucleolus
Hope this will help.