I currently have a single partition on the drive housing a Mint installation.
I want to install Ubuntu on this partition but I don't want to go through the hassle of backing up and restoring my private files in /home.
(Unlike Replace Mint with Ubuntu in dual booting system, I am not asking about the apparently missing option to wipe Mint alongside Windows and install Ubuntu just on that partition, OK?)
I understand during the install I can just select the partition as the root partition and not format it, correct?
That would mean that the installation would copy the files over the Mint files but would presumably keep my private files in /home intact.
Is this a viable option? Will it result in a usable Ubuntu installation or do I actually need to format the drive?
You could move /home to its own partition, then do a Something Else install, formatting only the / partition.
The wiki article here https://help.ubuntu.com/community/Partitioning/Home/Moving explains in painstaking detail how to move /home. I've done it successfully several times.
Most probably you have to format the partition. Selecting the root partition will also by default format the partition. Notice the "Format?" checkbox. Try unticking it if you want.
But why do you want to retain files from mint? This will be very messy and will probably result in things not working.