Authenticate with phone instead of password I understand that what I want is really strange.

We have something like shared build server (mac mini, yosemite), which is located in public place in office; basically everybody have physical access to it.

We have special user account created for our project needs, and all necessary tools / passwords etc are configured under this user account.

We have a rather big group of people, who should have physical access to this user account from time to time.

As a (predicted) result we have simple "password" which is known to everybody, even to people outside of this group.

Is there any way / third-party tool to use phone applications like Google Authenticator, which will allow access to this specific user account to a predefined group of people, without relying on just password?

In general I'm looking for any solution which might solve my problem - login to account with "public password" to specific people only.
I'm not aware of other 2FA softwares to unlock a Mac, however I've recently found this software that might be of use to you: http://www.knocktounlock.com/. It uses an iPhone to unlock a Mac. Otherwise, you could consider using an unused USB flash drive to serve as a key (for example Rohos Logon Key http://www.rohos.com/products/rohos-logon-free/).