This is a VERY touchy subject for me.
First (and the worst), there is an icon that I have witnessed weeping, that was in the old Blanco monastery in Texas. The priest admitted faking it, and I believe he was also involved in some type of sexual criminal investigation. It was a HORRIBLE tragedy. Also a horrible lie to do this. I don't know why he did it.
Now, I have been blessed in my life to have witnessed REAL weeping icons. Even ones that I got to hold in my hand that were not well known. They were at the time, scary, holy, mystifying, and beautiful. These were at monasteries as well, and one at a priest's home. I was holding and Icon of our Lord and savior Jesus Christ, where the hands and feet were secreting an oil. The priest wiped up the oil with cotton (for anointing). This icon WAS IN MY HANDS. I felt the wood on the back of it. Solid. No funny business, without doubt, drops seemed to "appear" not secrete, but slowly "grow" again out of the hands and feet. I will never forget this experience in my entire life. It is real that some icons do weep/secrete, or are mystically anointed somehow.
One other experience that I had was with a bronze cross (typical large priests cross). The priest discovered one Saturday evening that his main church cross (for veneration at the end of liturgy) was covered in a very strong scented oil. Well, he did not know how it got that way, and brought it to a sink and scrubbed it off with a sponge. He also used soap. At the time I was an altar boy, and I witnessed the priest scrubbing/washing the cross. He patted it dry, and we walked back into the church with the cross in his hand. It began GUSHING the sweet smelling oil out again, and completely was everywhere on his hand, sleeve, etc. I was sent to the local corner store to buy cotton balls!
The other icons were from very kind individuals that I would consider extremely credible, with nothing to gain. The most famous that I've seen was in Chicago, and the icon of the Theotokos weeped tears of oil. This was back in the 1980's. I had no chance to examine it at all. There are numerous accounts of icons on Mt. Athos weeping.
Now, to answer your question.
I've seen several and I can't fully answer your question, but I can tell you what others say.
Some say the weeping icons are because they "cry for the world".
Others say its to prove there is something to unbelievers.
Some say its to call back protestant Christians to Orthodoxy.
Some say its a flat miracle, end of story.
My personal opinion is that they are a miracle, and that they will mean something different and special to each person.