[The Might]

Whilst he walked in the wilderness for so long and fought for Rohan and Gondor it seems as if he did not have an own weapon, something he could in a way attach to. Usually in Tolkien's works heroes have a special bond to their weapons and they also have special powers. In Aragorn's case the shards of Narsil. The fact that he could not actually use the weapon is less important - it matters that THIS was his own weapon, the one he was bound to.

What he used during his journeys is rather irrelevant - bow and arrows, swords, knives, spears... whilst reading LotR one realises that he must have used most types of weapons considering his skill. So of course he had something to defend himself with, it's only that naming what it was was not important for Tolkien as no such special relationship as those mentioned above existed.