Interesting subject. Maybe the first questio nto ask would be 'Can an immaterial person have a detectable effect in the natural world ?'.
If we are talking about the Christian God, I do think that the answer to that question is yes. But the thing is, whatever effect he could have, it would be non-repeatable, and so all you would have is testimonies and personnal experiences.
Example: 'Jesus walked on water'. Now if that claim is a genuine miracle, whoever saw it would have seen it with his eyes and so with one of his physical senses. If it is not genuine, then it could just be a figment of the persons imagination. Either way, all you have is testimonies and personnal experiences, and in my opinion, that is always what you are going to get.
Now the question is, can you go from this 'subjective evidence', as RAZD calls them, and conclude a positive statement about the reality of the existence of an immaterial being ?
The only way I could see this question being answered by a 'yes' would be through an impecable 'police investigation' kind of process on a particular claim. How many eye-witnesss accounts of the event ? How reliable are these eye-witness accounts ? Is there a possible physical explanation to the event ? etc. etc.
All in all, I consider this to be very difficult to do for any given claim, and so I think that miracle-claim can only reinforce (rightly or wrongly) a pre-existing belief in the supernatural, and cannot necessarily be the starting point of such a belief.
On another note, this does not mean that I do not believe that the existence of God cannot be supported by objective facts, but you determined this as off-topic in your OP, so I won't go any further.