Alex is the first that comes to mind. I’m an author and playwright, and especially when writing plays, I prefer to make the characters ones who could be played by any student without making any changes. As a result, there are lots of characters who are animals or, if human, named Alex, Chris, etc.
I thought of Randy, though I think it is usually *Randy *for Males and *Randi *for Females.
Sidney.
Although during my lifetime it has transitioned from being a majority being males to a majority being females. So not sure it counts.
Kim was the first name that came to my mind, but I’m not sure why. Two of my coworkers are named Gabe – one male and one female.
Yes, Pat, but I think we’ve been primed to think that. 
Also, I think it depends on where you live. For example, Robin is a common girl’s name in the US (or at least not uncommon, these days), but not common for a boy. Perhaps in the UK it’s a common boy’s name.
Another vote for Jamie.
Jude. There’s Jude Law and then there’s my daughter-in-law.
I knew a girl named Timothy. There is Stoney, weird, right. John Wayne’s real name was Marion. I had an aunt Johnny.
Imma be honest here, when I drafted my OP, I expected someone would promptly post a YouTube link to the SNL sketch.
I feel … vaguely disappointed.
tip-tapping away by phone, but why would you care?
Loren for me and I have no idea why?
Alex
Dale.
God ,stop with Alex, I think it must be the winner!
This.
If you are from New York, Don and Dawn are nothing alike.
My nomination for a 50/50 name is Bryce/Brice. I see it equally for boys and girls, and both spellings pretty much divided.
Sue 
I do know a girl named Tomi, I know not usual but more common than Sue.
My kids are Cameron and Devon…boys? girls? or one of each?
Cam is the boy and Devon is the girl???
Pat