The problem here began long before the player ran out of Fate points, when he took an aspect to solely represent his father's sword and expected to use it on virtually every action just to represent the utility of having a sword. Fate Core, on the "Intro to Choosing Aspects" on page 36 says of choosing aspects:
Aspects which don’t help you tell a good story (by giving you success when you need it and by drawing you into danger and action
when the story needs it) aren’t doing their job. the aspects which
push you into conflict—and help you excel once you’re there—will
be among your best and most-used.
Aspects need to be both useful and dangerous—allowing you to
help shape the story and generating lots of fate points—and they
should never be boring. the best aspect suggests both ways to use
it and ways it can complicate your situation. aspects that cannot be
used for either of those are likely to be dull indeed.
Bottom line: if you want to maximize the power of your aspects,
maximize their interest.
My Father's Sword as an aspect is okay. There are potential situations in which that can be invoked or compelled to add interest to the story, but it is definitely on the weak side. It doesn't tell us anything about the character except that he owns a sword that was once his father's.
If the aspect had instead been something like I Will Avenge My Father or My Father's Legacy it would have a lot more presence to it and would enrich the story.
Invoking an aspect also needs to be justified in the story. You can't just say, "I invoke My Father's Sword for +2," every action. The player needs to describe how his aspect assists him when he invokes it. Lots of people have swords; what's so special about his?
Aspects (or just plain facts - what if I have a sword but we're not even considering it to be an aspect?) define narrative truths that can be significant advantages in the story. A man who has a sword can cut through ropes that an unarmed man has to untie, can intimidate unarmed people, can make cuts rather than bruises, and so on.
The mechanical benefits of aspects go above and beyond these narrative truths to provide a moment where the aspect is front and centre in the story, a focal moment in which it changes the course of fate.
Purely mechanical benefits are the purview of Extras and Stunts.