Linux - alternative places where to store pid file instead of /var/run

In short: you could store it anywhere (say, /tmp or /var/tmp), but /var/run is the preferred standard.

/var/run is the Filesystem Hierarchy Standard:

This directory contains system information data describing the system since it was booted. Files under this directory must be cleared (removed or truncated as appropriate) at the beginning of the boot process. Programs may have a subdirectory of /var/run; this is encouraged for programs that use more than one run-time file.[footnote 37]

And a desirable feature is that most distros clean it automatically (unlike /tmp which is not cleaned upon boot in some distros) - this avoid stale pid files:

The normal location for pidfiles is /var/run. Most unices will clean this directory on boot; under Ubuntu this is achieved by /var/run an in-memory filesystem (tmpfs).

It's your choice where to store it, but I would go with the standard.

If you don't have access to /var/run, you should store the pid file in the user's home directory, e.g. ~/.my_app.pid.

If it's non-root write access to /var/run that you need, then note that there are subdirectories in /var/run/user for individual users. You just need to get the UID of the current user:

/var/run/user/[$uid]