Here is a counterexample, which is probably not "nice". Let $X$ be the Warsaw-circle. Let $X_n$ be the obtained from the Warsaw-circle by thickening the limit inverval by $1/n$. The intersection of all the $X_n$'s is the Warsaw-circle, and its first homology vanishes.
Each $X_n$ is homotopy equivalent to $S^1$ and the inclusion $X_{n+1}\rightarrow X_n$ is a homotopy equivalence. Thus $\lim H_1(X_i)$ is $\mathbb{Z}$ and there cannot be a surjection.
Meta: Every $X_i$ is has the structure of a compact CW-complex. However the intersection behaves badly, we cannot arrange $X_{n+1}$ to be a subcomplex of $X_n$ in this example; otherwise the intersection would be a CW-complex, which it is not. I guess this has to be a part of the niceness condition. But then on the other hand, since all spaces are assumed to be compact, we have $X_i=X_{i+1}$ almost always and thus both inverse systems stabilize. I have no clue what a good niceness condition could be.