When triggering profiler it would be very convinient if we could also specify output filename using request variables.

It could solve at least two problems:
1) main problem: if there are already some profiles in the output folder before profiling, you can get lost among previous profiles. But when you explicitly sepcify the file-name, you know what file to open then.
2) it could theoretically ease automating the profiling process (for example, a simple macros for manual profile analysis: launch the profiler and then auto-open the profile). For example, such macros could theoretically be implemented inside IDE's (like there is now Run/Debug buttons in PhpStorm, there could another button be added - "Profile", which would auto-open the generated profile).

*) I know that XDebug adds a response-header telling about the generated filename, which actually solves the problems above, but that requires somewhat analyzing it [response], so adding a new request variable for the filename could ease the process.
**) For securiy reasons, I think it would be wiser to allow only spefifying additional param for "profiler_output_name", not the whole filename (I mean specifying an additional "%"-placeholder). In this case one could strictly forbid getting filenames from request by just not adding this placeholder to the pattern, no additional configuration variables are needed.