The thing about a Rider's sword is that via the ancient magic of the elves, the sword is unbreakable, always sharp, and can hold magic reserves; all needs of a Dragon Rider Having grown accustomed to such a sword, other weapons often fail him, breaking from the force of his blows (Eragon has super-strength due to becoming part elf), and dulling from his poor technique.
He tries a magically reinforced staff, and a similarly reinforced blade. He uses magic to make his fists studded weapons, which works out well, but up to the point I've read, he has mentioned it often and repeatedly, and it's clear it's one of his main goals at the moment.
