I name my chicks based on their personality or purpose. If I'm looking for eggs, not meat, then names like 'omelet', 'cake', 'sunny (side up)', 'scrambled' come to mind. If my ultimate goal is to eat the meat should they turn out to be cockerels then I look at names like 'Kay-Ef-Cee (Kentucky Fried Chicken)', 'Fricassee', 'A la King', etc. Some I have named after their looks, I once had a rooster named 'Fancy Pants' because he was my first foot-feathered bird. I once had a rooster named 'Yahoo' because he didn't cock-a-doodle-do... he YAHOO'ed!
Deciding the purpose of your flock, be it eggs, meat or pets, will make the naming process easier.
I usually name my chickens based, at least in part, on their country of origin or a favorite character.
My Speckled Sussex, a British breed, got a British name, Bronwyn, that's also the name of a character from a movie I love.
Eleanor, a Maran, is named for the French queen, Eleanor of Aquitaine, a very brave ruler.
The Buckeyes, all-American girls, are named for three TV characters: Althea, Beatrix and Cordelia.
Once in a while, I break down and name them for physical features. Honey is a golden beautiful Buff Orpington, and one of my Cochin Bantams is named Gabby because she talks all the time. My roo, an Iowa Blue, is named Sir Henry the Loud because he is.
Good luck with choosing names. I've never had to deal with a mongoose, but I have had trouble with minks. Burying hardware cloth and wrapping the runs in it, not chicken wire, may help.
Welcome to BackYard Chickens. I hope you get the answers you need.
I usually go on Pinterest and search “names with meanings” and a whole bunch of beautiful and unusual names come up. Then I pick one with a meaning that I like, for instance my chick Deloise, her name means fighter because she almost died right after hatching but didn’t give up, now she’s a healthy happy chick and then there’s Sidra, who’s name has to do with stars. I just really like that name! Another thing I’ll do is name them after my favorite book characters or after characters who have pretty names, I read A LOT. Nessa’s (my pullet) name is from the Silmarillion (a Tolkien book, if you don’t know) and Cressida and Octavia’s (although I might need to change it to Octavian, we’ll see) are from The Hunger Games (they’re ducks).