How can I have "names" for my background processes (not just the command)?

I run a lot of vim windows, and often put them in the bg. The problem is that they all say vim when I list jobs so it's hard to distinguish.

- suspended  nvim
  - suspended  nvim
  + suspended  nvim

One way around this is to open a specific file when starting vim:

nvim ./myfile

jobs:

+ suspended  nvim myfile

But this is a poor substitute for a name. Another option is to (mis)use env vars like this to name the buffer:

vim name=my_name

But the output of jobs could be more usable, and has the disadvantage of needing to preparing for the possibly unneeded backgrounding of the process at the moment of staring the process.

- suspended  nvim name=my_name
  + suspended  nvim name=other_name

Ideally I'd like to be asked to give the process an optional name when I press ctl-z. Perhaps there's a plugin out there that handles this, but I haven't found it. I would consider writing it myself, but I wouldn't know where to start. Presumably by re-mapping ctl-z to some other program? And maybe I'd have to write something that sites in front of fg and jobs as well to control the output... Any suggestions for this?