The Original Movie script (post adaptation) sheds some light on this:
I believed that all I had to do was
point my finger and anoint whoever
I chose. I was wrong, Neo.
Terribly wrong. Not a day or night
passes that I do not think of them.
After the fifth, I lost my way. I
doubted everything the Oracle had
said. I doubted myself.
And then I saw you, on the Net, searching for me and everything
changed. I felt this charge, this electricity through my whole body. I
started laughing and tears poured out of my eyes, because I knew, I
knew that you were the One.
As you allude to in the phrasing of your question, Morpheus doesn't know who the one is, but he believes it to be Neo; and for Morpheus, belief is everything. It can move Mountains.
Interestingly, Neo is not the first 'One' that Morpheus has discovered, and there have been at least 5 other candidates of his selection:
CYPHER I'm going to let you in on a little secret here. Now don't tell
him I told you this, but this ain't the first time Morpheus thought he
found the One.
NEO Really?
CYPHER You bet your ass. It keeps him going. Maybe it keeps all of us
going.
NEO How many were there?
CYPHER Five. Since I've been here.
NEO What happened to them?
CYPHER Dead. All dead.
NEO How?
CYPHER Honestly. Morpheus. He got them all amped up believing in
bullshit. I watched each of them take on an agent and I watched each
of them die. Little piece of advice: you see an agent, you do what we
do; run. Run your ass off.
It seems that Morpheus method of choice for initiating potential candidates is to remove the possibility of doubt that they may not be the one, and supplant it with absolute faith. Perhaps, as is proven to be true, The One cannot manifest their true potential unless they undergo 'a revelation', and this is only possible without doubt.
This blind faith could perhaps be a subterfuge by Morpheus to encourage The One, which would go some way to explaining why he appears shocked, dumbfounded even, when it emerges that Neo is infact The One.
