We can do the basic browser troubleshooting steps to see if you're just experiencing random browser issues.
We can start by signing in to your QBO company using a private window. This step will disable the extensions and stops the cache from saving your search history.
Google Chrome: Ctrl + Shift + N.
Safari: Command + Shift + N.
MS Edge and Firefox: Ctrl + Shift + P.
If you can select the employee in the Timesheet section now, you can clear your browser's cache. If not, please use a different browser to see if this only happens to the one you are using.
Let me know if you have any other concerns with your employee's profile. Take care and have a good one!