I hear a lot about how weapons training is important for learning the fundamentals of aikido, but I've yet to train with an instructor who satisfactorily explains why or how.
I've trained with teachers who successfully teach aiki-weapons in relation to empty-hand technique: a) from a technical perspective, as in this empty-hand technique or this footwork can be mirrored with the bokken, etc., and/or b) by explaining issues of timing, centerline, tegatana, mental "connection", etc.
The perception of the usefulness of aiki-weapons as practiced today will depend on your definition of "fundamentals of aikido". The way I see it and circular logic notwithstanding, I still don't understand what makes any of the prominent aiki-weapons work out there "aiki". That said, I don't do aiki-weapons as I think a lot of that training is counter-productive to my work on aikido fundamentals as I understand them.
And don't get me wrong, I like and do (cross-train) weapons. Weirdly enough, the "outside" weapons work I've done have helped my aikido far more than any aiki-weapons I've been exposed to. I've also seen this phenomenon on friends who cross-train in classical Japanese and FMA weapon systems.