I'm also stumped. Apart from Borderlands, I likely will not get any characters in any games made after the SNES era (with the exception of a few hyperpopular PS1/GBA games)
Mine was from Hotline Miami, an indie game on PC. If you haven't played it then I recommend picking it up on Steam. You can pick it up for about £2 (about $3.30-ish?) when it's on sale and it's great fun.
He doesn't have a name. I would have accepted anything providing it contained the word 'beard'. These include, but are not limited to, 'beard man', 'beard face' or 'beardy guy from Hotline Miami'.