I'd like to move my BOINC installation, and the progress on it, to a new installation on another hard drive.

I've reinstalled my OS, with the same user name, and computer name, on another hard drive, as I think my current drive may be failing.

I'd like my new installation to be recognized as the same computer by the servers, and be credited as such, and the chart of the host average and host total in BOINC Manager to continue from where the other installation left off.

BOINC Manager says it is version 7.2.42 (x86).
I'm running it on Linux Mint 17.3 32-bit Cinnamon.

I do not recall how I installed it. I managed to get it installed, and promptly forgot just about everything I did to accomplish that.

The work units appear to be running under a different user name than my own, named "boinc".

A thread about a Windows XP installation
https://boinc.berkeley.edu/dev/forum_thread.php?id=7150
basically said that you could just move the data directory to the new drive, and then reinstall BOINC.
Is it similar on Linux?

How do I find out how BOINC is presently installed? How do I move the relevant parts of my present installation to the other drive?

You probably installed it with package manager (Synaptic). You can check with the package manager if "boinc-client" is installed, or you can check if the client is running as service with "service boinc-client status" or can check if "/var/lib/boinc-client" exists.

Assuming it was installed with package manager then:

Copying BOINC's data directory to the new disk is easiest to do before you install BOINC on the new OS.

First, boot to the old OS.
Uninstall BOINC (boinc-client) using package manager.
Boot to the new OS.
Open file manager and make sure the old disk is mounted (has eject sign next to it).
Open Terminal.
Run the following commands:

sudo cp -a /media/your-account/partition-name/var/lib/boinc-client /var/lib/boinc-client
sudo chown -hR boinc:boinc /var/lib/boinc-client

Install BOINC with package manager.

I haven't actually tested these instructions. Proceed with caution.

Uninstalling BOINC from the old OS is so that if you someday need to boot into it BOINC will not run there. If it does it'll just make a mess.

I checked in Synaptic, for "boinc", and it shows "boinc-client", "boinc-manager", and "libboinc7" are installed. I suppose I did install it using the package manager. I wasn't sure about that, I know I had seen packages online, thought I may have used one of those. Probably not.

The directory "/var/lib/boinc-client" does exist.

As for being careful, I'm not entirely sure how to be careful, in this case.

The command "service boinc-client status" brought up:

 * Status of BOINC core client: running
 * Scheduling of BOINC core client: 1558
pid 1558's current scheduling policy: SCHED_OTHER
pid 1558's current scheduling priority: 0
 * Scheduling of BOINC core client's children: 28222 28290 
pid 28222's current scheduling policy: SCHED_BATCH
pid 28222's current scheduling priority: 0
pid 28290's current scheduling policy: SCHED_BATCH
pid 28290's current scheduling priority: 0
 * OOM killer status for BOINC core client:
PID 1558: adj 0
, score 
1

PID 28222: adj 0
, score 
5

PID 28290: adj 0
, score 
7

Ok, I think you can proceed with the instructions. You of course need to replace "your-account" with the name of your account and "partition-name" with the name of the partition holding the old OS. The partition name may be some long string of numbers and letters.

I have attempted to follow the instructions given, but have hit a snag. I have stopped at this point.
When I ran the command

sudo chown -hR boinc:boinc /var/lib/boinc-client

I got the response

chown: invalid user: ‘boinc:boinc’

and I have not proceeded past that point.

What should I do now?

That's what happens with untested instructions. Because BOINC isn't installed yet you don't have boinc user or group.

Run these commands to add the accounts and then continue with chown and installing BOINC:

sudo addgroup --system boinc
sudo adduser --system --ingroup boinc --home /var/lib/boinc-client --gecos "BOINC core client" boinc

Probably the easiest way to create the boinc group and user - the usermod just tidies upa description field

sudo adduser --system --group boinc  --home /var/lib/boinc-client/
sudo usermod  --comment "BOINC Core user" boinc

This should create the home directory and the user / group records correctly, although the install normally does that.
edit: Juha beat me to it! You may need to add boinc to the video group if you plan to use the GPU for crunching

boinc-client install script takes care of adding boinc to video group.

Thanks, Juha and Agentb!
I used Agentb's command to add the user, and now the chown command works.
After installing BOINC, it shows my projects, and progress, in the manager.
It all seems to be just as I wanted it.