Bill, Betty, Rich, and Kate - none of your answers are (is? help, Laurie) correct.
Hint #2: The answer comes very early in a book I bet all of you have.
That was a secert meeting Roger. No one is supposed to know about it.......
Gene, you're making me laugh again!
Hint #3: The handshake and brief conversation occurred at a public reception.
All I can think of is Boston Corbett because he met Lincoln at a reception when his NY unit arrived in D.C. at the onset in 1861, to guard the city from a suspected invasion from Virginia. However, I don't think that Booth knew Corbett
I'm trying to think of a character that would socially priviledged enough to attend a public reception with the President and the only one I can come up with is Mudd.
If this is a side character in the assassination story, but not directly related to the assassination, then I'm going to vote for John Ellsler, one of Booth's partners in the oil fields. Ellsler was into the theater business, was well-respected, and was the manager of a musical enterprise in Cleveland (I think) that might have put him in the presence of Lincoln.
This is a humdinger, Roger. Can I buy the rights to this one? I could use this.
Careful, Roger. He's going to use it against us Southerners.
Do the right thing Norton. Or the cheese lady loses your address.
Threats! Do y'all see that??? Threats from the Yankee supplier of the cheese!
John Mathews (if it was a side character)?
All guesses continue to be wrong.
Hint #4: He is not a "side character" such as Ellsler. He is an extremely familiar name to anyone who has read Lincoln assassination books.