Can I update the operating system on a Mac mini with 10.9.5 OSX?

Upgrading a macOS depends upon the model Mac, specifications, and year. To get more information about your computer, choose:  (Apple menu) > 'About This Mac' in the upper left corner of any window, then "More Info..." or "System Report". There's more about this in "About System Information [Profiler] on your Mac" - https://support.apple.com/HT203001

Use the computer information, including the model year, and the links below to determine what is possible with your computer.

Click on this link to read how to get different versions of macOS --> https://support.apple.com/HT211683

- The article also has links to system requirements for each OS version which you should check against your exact computer model. Note some of the model identifier articles are outdated. For any model produced in the last 5 years or so you should use the system requirements links for each OS version instead. This web site also has a reliable presentation of model and OS version compatibility: https://eshop.macsales.com/guides/Mac_OS_X_Compatibility

- It helps to have more RAM (memory) than what is noted as "minimum" for any particular system or your computer may run slowly.

- You may find to get links for High Sierra and later using the Mac App Store application to open a functioning page you need to use Safari.

- Sierra, El Capitan, and Yosemite download as a .pkg. Click on this to have a converted version placed in your Applications folder as an "Install OS..." application. Run the installer and follow the directions.

- Upgrades may take some time (up to an hour or more) to install, with restarts and blank screens.

Before upgrading, back up at least your data in case something goes wrong. See this support document: How to back up your Mac - https://support.apple.com/mac-backup

Check if your old apps will still work with with the OS you want to install - https://roaringapps.com/apps This is particularly critical if you are upgrading to Catalina or newer, as older 32 bit apps will no longer work. To check for those, try this 32 bit application scanner application --> https://www.stclairsoft.com/Go64/

Which exact model is your Mac mini?

Back up your data, check your applications for compatibility, and click on El Capitan or High Sierra.

(202568)

The mini model can be upgraded up to OS X 10.13.6 (High Sierra)

Ref: Get macOS High Sierra - Apple App Store

You can upgrade to the OS versions indicated by Niel but you have very little RAM (2 GB) for those system versions, so expect things to run slowly.