The Whitehead link fibers with a twice-punctured genus 1 fiber. Since the Seifert-Webber Space is a cyclic 5-fold covering of the Whitehead link, the image of Whitehead link in this cover should again be a fibered link with a twice-punctured genus 1 fiber. So this will give a genus 3 Heegaard splitting.
I reckon this argument should work to show each of the two isometry classes of 5-fold cyclic covers (as mentioned in the comments to this question) have genus 3 Heegaard splittings.
Edit: Of course Agol's comment is right. I wasn't considering the other possible 5-fold cyclic branched covers.
So if I've got my head on straight about it, these other covers should be dual to a thrice-punctured genus 1 fiber of the link exterior. (Such a fiber is obtained from adding the twice-punctured torus Seifert surface with a suitably oriented once punctured torus bounded by one component and disjoint from the other.) Then in the cover, as well as in S^3, the two boundary components of the fiber on the same link component join together to make a non-orientable surface with $\chi = -3$ and one boundary component. The boundary of a neighborhood of this surface then gives a genus 4 Heegaard splitting. Presumably, this splitting is equivalent to the ones Bruno obtained.