'I' might be thought to be a convenient shorthand word to refer to all of the intimately inter-related properties which make up 'me' - memory, hair, fingernails etc. I suggest that in fact, rather than constituting all of these things, it is none of them. We may use it, in this way, colloquially but reflect for a moment. I know that 'I' have always existed for the duration of the lifetime of this body. The feeling of 'I' is unchanged despite the fact that the body has multiplied in weight dramatically since birth, changed all of its cells many times - I cannot be this body. What is my (colloquial) mind but the thoughts, opinions, beliefs, memories, etc. - which have changed drastically and continue to change - I cannot be the mind. All of the 'properties, functions, qualities' that one might think make up 'I' are merely ways in which I limit myself by attaching to some aspect of this body-mind entity.
Part of the problem lies in thinking that the word must be 'a thing in itself'. It is not a 'thing' at all. Its everyday usage is another part of the problem. In using it to refer to one or more or all of these properties, we are really talking about the ego, which does not exist at all. The 'real' I is pure unattached awareness, singular and without limiting properties. We (the person) are continually looking outside of ourselves for a justification for our existence, a 'meaning' for our petty lives. This is all pointless. The 'person' is a fiction, a 'mask' as Alan Watts points out, the megaphone through which the actor projects his voice in the role he plays. Moreover, he exists for but an instant. The truth is found not by looking to the world, searching for happiness in external objects, but by looking inwards to find the real 'I' behind and beyond all of this.
As Death says in the Kathopanishad: -
"God made sense turn outward, man therefore looks outward, not into himself. Now and again a daring soul, desiring immortality, has looked back and found himself.
The ignorant man runs after pleasure, sinks into the entanglements of death; but the wise man, seeking the undying, does not run among things that die.
He through whom we see, taste, smell, feel, hear, enjoy, knows everything. He is that Self.
The wise man by meditating upon the self-dependent, all-pervading Self, understands waking and sleeping and goes beyond sorrow."