The short answer
Yes. Unlike most lines in this collection, this one traces to Mandela himself, and to more than one occasion. It is most often sourced to his address at the launch of the Mindset Network in South Africa in 2003, where he said education is the most powerful weapon we can use to change the world — adding that the Mindset Network was part of that world-changing arsenal.
He had said it before
The Oxford Treasury of Sayings and Quotations traces the sentiment to a speech at Madison Park High School in Boston on 23 June 1990, during Mandela's visit to the United States months after his release from prison. Contemporary reporting of that speech renders the line in several forms — one American network reported it as a weapon to prepare young people for their role as leaders of tomorrow — which is exactly what happens when a speech is quoted from the room rather than from a text.
The one real discrepancy: we or you
The version printed on posters uses you: the weapon which you can use. The best-documented deliveries use we. It is a small word carrying a real difference — we makes education a collective project, you makes it a personal instrument. Both are defensible renderings of what he said on different days; we simply note which is which rather than pretending the wording is fixed.
Why publishing a Verified result matters
A section that only ever debunks becomes a machine for cynicism, and cynicism is not the goal. The point of checking is not to discover that everything is fake — it is to know which things are real. This one is real, and it is stronger for having been checked.
What this means for your English
Look at the relative pronoun: the weapon which you can use. Many English speakers would say that here, or drop it entirely — the weapon you can use. All three are correct; which is the most formal, and dropping the pronoun is the most conversational. Being able to move along that scale deliberately is a genuinely advanced skill.