Well, if you’re asking for the first one that comes to mind, then it’s Robin. Having read through the answers, I think there are some answers that are more thoroughly unisex (Jamie is probably the best one, Alex, Taylor, and Jordan are all good ones), but the first evenly unisex name to pop into my mind was Robin.
I have a friend who named her son Dylan, which I think occasionally gets used for girls as well.
Robin was the first one that came into my head, and as has been mentioned I don’t think of things like Pat, Chris, or Alex truly unisex in the same way that Robin is. While it might not be a popular boy’s name any more, there are a lot of famous male Robins.
I’ll also note that I went to a HS that at one point had both a boy and a girl named Morgan. But I only remembered that after someone else mentioned the name.
Additionally, both of my parents’ middle names were originally Lynn. It definitely ended up as my middle name as well, well after the time period that Lynn as a boy’s name was even close to popular.
Yeah, Pat was the first one that popped into my brain. Most definitely because of SNL.
Unisex, not Palindromes.
I know both men and women who have gone by ‘Bob’, short for Robert or Roberta.
But it was kind of a tongue in cheek reply…
My thought was Taylor. But Chris is a better choice.
Not to this NE person. Don=cot, Dawn=caught
Aaron=parrot Erin=penny
Chris/Kris
It’s the common nickname for a large number of gendered names. If I just heard it, I’d have very little idea of which gender the person was. If I saw it spelled, I would think Chris with a “Ch” is slightly more likely to be male, while Kris with a “K” is slightly more likely to be female, but there are enough of both genders with both spellings that it’s pretty hard to tell.
After that, Pat, Alex, Jess(i)e, Jaime, .
A lot of “boys’ names” have become “girls’ names,” so if I know how old the “Jordan” or “Robin” or “Taylor” in question are, I don’t think of the names as unisex.
Evelyn ( with a long eee sound) used to a mans name. We have used the name Lynn for both sexes is our family.
Chris. It’s my wife’s name. Though, hers is short for “Christiane,” which is far less common than Christine / Christopher.
For that matter, Shirley used to be a man’s name. Charlotte Bronte wrote a book titled Shirley in 1849 about a girl who’d been given a boy’s name, because her father really wanted a boy when she was born, and how this affected her. It produced a fad for naming girls “Shirley,” and once Shirley Temple became famous, it was never a man’s name again. Shirley Temple turned it from a mostly girl’s name that sounded kinda tomboyish, and was still sometimes given to boys (like Robin in the US now), to a really girly-girlish name the no one would ever give their son, any more than they’d name a son “Melinda,” or “Sarah.”
The memoirist and journalist Vera Brittain is the mother of Shirley Williams, MP (and Baroness), and when Brittain describes her daughter’s birth, she says she named her after the Bronte character, because she wanted her to have a “scrappy” sort of name to live up to.
Williams was born in 1930, and Temple made her first vehicle in 1934.
Another vote for Jordan. I think my acquaintances named Jordan are split exactly 50/50.
Then there’s Alexis. In an English-speaking country, I think girl, anybody else I’d assume boy first.
Shirley is the name of one of Anne and Gilbert’s sons in the Anne of Green Gables series. (Anne’s maiden name was Shirley.)
As for Evelyn being both a man’s and a woman’s name, that’s a clue in the following mystery…
Mrs. McGinty’s Dead, by Agatha Christie
When I was at school it was about 50/50. The most common male name was David (you can date me from that statistic if you want), but the total Chris’s outnumbered the David’s.
I just thought of another couple. Jody and Toby.
I think Jody was trending towards girls, but then Family Affair gave pop culture a boy Jody.
I know several Jewish women who go by Toby. Their full name is Tovah, and in Hebrew the letter that makes the V sound is a letter that can make a V or a B sound, depending on where it is in the word, and what the following or preceding letter is. “Tovah” in Hebrew is the feminine form of the word “good.”
I don’t think I’ve known I gentile woman named Toby, but I really have known a lot of Jewish women called that, and the number of gentile men I’ve known named Toby is pretty few. I don’t know a Jewish man called that. I think I’ve actually known more boy dogs named Toby than boy humans. I did go to intermediate school with a kid whose parents had unfathomably named him “Toby Dick.” (Dick was his last name.)
Sandy!
That doesn’t make it not count. Even if it weren’t a unisex comical character, it’d still be my go-to…I know dozens of female Pats and at least six male Pats. Never met a female Kelly or a male Leslie.
But that’s just me. I’m sure a lot of our Dopers can tell you their first grade teacher was a man named Ashley, since it was pretty common in 1903.
Interesting! I can remember one male Kelly and many women named Kelly.
Cheers: Kelly, Kelly, Kelly, Kelly…
I’ve learned something today! 
One of my favorite musicians is a woman named Toby Lightman. I’d always wondered about her name, but I do know that she’s Jewish, so it now makes sense!
Homo-phonically, Joe and Jo.