The "Append preset name to file name" checkbox in Preferences is one way to do this, but you could also use an Ingest Preset to access more robust renaming options:
Create a new Ingest Preset (Preset > Create Ingest Preset...)
In the Ingest Preset Settings dialog, turn on "Transcode files to destination" and choose a destination path for your output.
Select an Encoding Preset from the Format & Preset menus.
Turn on "Rename Files" and choose "New Preset" from the popup menu
Click the "Add Preset Field" button and choose "File Name"
Click the "Add Preset Field" button again and choose "Custom Text".
Type a custom string in the text field (ex: "_youtube HD")
Click the Save button and type a name for your renaming preset.
It looks like using the ingest method doesn't get the "append preset name to file name" option despite it being checked in my preferences. And there isn't a field for it in the ingest method (only Custom text/Date/Time/File Name). Is there a way to get the preset name added to that list? Or am I missing something?
Also, Is there a way to get a relative path option for the destination (like ../export/) instead of an absolute path?
Unfortunately ingest presets don't honor the "Append preset name to file name" preference, nor is there a relative path option when choosing a location. Both are good features requests tho'!
The User Voice portal is the best place to add suggestions like these, go to Help > Provide Feedback... from any Adobe app to access the site.