The security of the sponge construction rely on two parts:
- the size of the capacity.
- and the strength of the permutation used in the construction.
This permutation is expected to have at least the following requirement:
- provide a strong diffusion (in Keccak this is provided by $\rho$ and $\pi$).
- provide confusion ($\theta$ and $\chi$).
In the case of Keccak, $\theta$ is an operation mainly columns oriented.
Which is why $\pi$ will ensure that every bits of a column is evenly spread in the slice. This prevent the creation of patterns.
$\chi$ is the main ingredient in Keccak-$f$. It is the only part that is not linear. Without it Keccak would be super weak to cryptanalysis.
propagation of a difference through $\chi$
Lastly, Keccak-$f$ provides a weak alignment (resistance to truncated differential cryptanalysis). The idea is to make sure that the differences are not constrained by a subdivision of the state (byte for AES or a group of 5 bits in the case of Keccak). However, due to the weak alignment of Keccak-f, finding the lower security bounds of the algorithm is harder.
If another permutation provide such characteristics (NORX permutation? I'll let Richie Frame answer that part. He loves NORX). Then I guess it would be another decent choice.
I haven't studied LFSR.
TL;DR: The candidate permutation must provide strong diffusion, confusion and if possible a weak alignment.