I'm a developer, trying to land a job at an exchange. What type of software would an exchange need? Want to build something to use as a resume I'd like to give a good impression by solving a problem they already have, but I'm not sure what could it be.

Thanks!
I think there are few values that every exchange is looking for. 


ability to work with multiple coins (exchanges = shitcoin casinos)
ability to generate the addresses and manage them securely on-scale within organization
knowledge of API interfaces 


Regarding second point (as it's a bitcoin-related board) I'd advise looking at few projects that you can work on yourself, copying the ideas:


Key derivation tool: https://iancoleman.io/bip39/
Bitcoin transaction fee estimate like: https://billfodl.com/pages/bitcoinfees or https://bitcoinfees.earn.com/
Maybe some Lightning Network related project if you know that the exchange you are applying in may be interested


Although it relates to bigger exchanges only, as smaller ones use external order books and usually work only on the frontend.