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In The Matrix (1999), Morpheus says to Neo:
Neo: You know something. What you know, you can't explain. But you feel it. You felt it your entire life: Something's wrong with the world. You don't know what, but it's there. Like a splinter in your mind... ...driving you mad.
What does this quote mean: Like a splinter in your mind?
A splinter is a tiny piece of wood that has pierced the skin and is stuck under the top layer of skin. It's generally not serious to get a splinter but it is annoyingly painful. Splinters can be difficult to extract, and they also can work themselves in deeper.
So, a splinter in one's mind is a small, mostly harmless but irritating thing that cannot be removed easily and continues to nag and bother you. Sometimes you are distracted from it and you forget it's there, but then you have a free moment or it is bumped and you are fully aware of it again.
Adding to the existing answer (which I think better answers the actual question), the quote might have been referencing the novelized sequel to Star Wars (later Star Wars: A New Hope), Splinter of the Mind's Eye.
Since the Matrix was explicitly built by the machines to reflect human mindsets, it's likely that the pulled in aspects of popular human culture to reflect the harvested human virtual experience. Thus, Morpheus would have references to prominent bits of human pop culture like the novel that was intended as a possible sequel to the first Star Wars film.
