Some of those suggestions verging on the ridiculous side are Capy and Bara, Jono and Ben, Boris (Johnson) and Donald (Trump), and Nacho and Burrito.
When thinking about names, primate team leader Amy Robbins said to "think about their personalities - they're feisty, full of life, have big personalities for their size and bear in mind their South American origin".
The winners will be announced on Monday, July 4. They will be awarded zoo passes for their efforts.
If you have a name that's fitting for these little critters, make sure entries are in by 12pm Friday.