I’d probably go with a pink tongue skink. My parents used to have one, until some weaselly little shit stole it. Arorable lizard.
Baby tortoises such as the one shown below.
I think mice are the cutest things. They would be my favorite grown animal that fits in my hand. My favorite baby would be a dachshund puppy.
As an award show accessory. To add objection to absurdity, he named it “Johnson” and proceeded to ask female interviewers if they wanted to touch his Johnson all night. LOLOL!
Sixthing kittens, but rats are pretty great too.
'Nother vote for hedgehog! They’re only slightly prickly.
I don’t like small dart-y animals for the most part, so most of my favorites are birds or cold-blooded: turtles, lizards, fish etc. As for mammals, I think the smallest adult ones I like are chipmunks.
I have noticed a distinct lack of spiders in this thread. They can easily fit in most people’s palms…except for the huge bird eating ones, that is. 
They do, but if they’re comfortable with you they keep them flat against their body and they won’t poke you. And the spines are sharp but not needle sharp, so even when they’re curled up into a spiny ball if you’re careful you can still hold them and it doesn’t hurt.
Is Sonic the Hedgehog hedgehog-sized? I know he wears shoes and stuff but I bet they are hedgehog-sized shoes.
If he’s really hedgehog-sized, then I choose him.
That wikipedia photo is exactly - EXACTLY - how Gollum/Sméagol looked in my mind when I was reading The Hobbit as a kid! Y’know, only bigger.
I’ve got to give my vote to the hedgehog. I’m tempted to vote for that Japanese dwarf hamster, but if I ever got hold of one, I’d ignore every other thing in my life, and just eventually expire of cute. So I’d better stick with the hedgehog - cute, but not lifethreatingly so.
Bunnies!
…Or a very, very tiny hippo. 
Cool looking, yes. Cute? If it floats your boat. I don’t think they take to a terrestrial environment though. They’re so stingy, it’s like thousands of little nematocyst hugs!
Seconding (thirding, whatever) rats, sugar gliders, and bearded dragon (although those are pushing the “palm” envelope).
He said it had to fit in your hand, not draped across it… 
Baby guinea pigs. I know they fit from personal experience.
Hamster!
Maybe yours fits.