I always start by setting my zoos in a specific location and I try to choose a realistic but fictional name based on that place whether it’s Finland, Australia or Japan as examples.
In North America especially there are lots of zoos in parks so say your name is, for example Smith, you could call it Smith Park Zoo which sounds realistic and has a nice personal connection if based on your name. Lots of places have natural features in their names like Cedar Falls or Westlake or Pine Valley. Cedar Park Zoo, Pine Park Zoo etc. There are so many Springfields so that could be one to pick.
Look around where you live, is there a street name or a neighbourhood whose name you could use? Say you lived on Hillfield Road, you could have Hillfield Zoo, Hillfield Wildlife Park, Hillfield Park Zoo.
Or combine two names. My Japanese zoo is called Matsuda City Zoo. It sounds realistic, has a meaning - the characters stand for pine and field - and it’s actually a combination of Matsumoto City and Iida City which are both cities in Nagano in central Japan. Looking randomly at a map of Toronto I can see places called Rexdale and Smithfield. So you could have Smithdale Zoo or Rexfield Zoo and they sound realistic. Are there place names near where you live that could be combined like that?
You could name it after a family member you like or a friend or even a fictional character. Say you like Harry Potter, there’s already a Potter Park Zoo in real life but you could have Potter Falls Zoo, just as an example. Or Granger Park Zoo. You could also name a zoo after a town in a book or film like Avonlea or Cabot Cove.
There are lists of fictional place names online which could help. There are also fictional place name generators online.
I also sometimes look through an atlas or on google maps and find the name of a tiny village or a lake somewhere and just use that.
There are so many possibilities of names and ways to make it meaningful to you so please don’t be discouraged.