Well a large part of why Gordon Freeman is so popular is not atually anything to do with him per se- it's because he's the protagonist of what is, let's be honest here, the best fucking FPS series ever made. I mean, look at the character he's constantly being compared with in this thread. Master Chief isn't popular because he's a great character, because he isn't. He is, if anything, more boring and generic than Freeman (I'll explain how that is later) but he's become one of the most recognised game characters of this generations simply because he's the protagonist of Halo, one of the other most popular FPS series ever. Games like Modern Warfare can't do this because they don't have a fixed protagonist, but Chief and Gordon are fixtures in their series.
So what makes Gordon more popular than Master Chief (other than Half-Life being better than Halo) even though he has NO personality at all? It all goes back to the feature which first made the original Half-Life memorable- its total immersion storytelling style. The game does not use prerendered, 3rd person cutscenes, or remove control of Gordon from you at ANY point (with the only exceptions being when Gordon is physically immobolized, such as when the soldiers capture you in Half-Life or when you're buried in rubble at the start of Episode 1). You can always at least look around. You can stand in front of characters when they're talking to you and listen, or you can wander around looking at the walls and letting them prattle on in the background. To make a long theory short, in Half-Life you ARE Gordon Freeman. The effect of the silent protagonist has NEVER been used so well, not in Zelda, not in Chrono Trigger, not anywhere. This is where Master Chief fails- he has his OWN character and personality, preventing me from overlaying mine on him, but his is completely boring and generic. When I was playing Half-Life 2 I identified so well with Gordon that I intently roleplayed him at every opportunity, nodding or shaking my head with the mouse when people asked me questions, looking away awkwardly when Eli teased me and Alyx and even talking back to the characters out loud. Gordon may not have a voice or personality, but that's because he has mine.
Finally, there's the fact that he's such an original character, but his story develops so well. As has been pointed out by countless people, Gordon is a theoretical physicist who gets thrown into the biggest shit-fest in human history by sheer chance and comes out the other side alive and covered with alien gore. Then he finds himself on a dystopian earth where he's become a messianic figure to the human resistance and he helps topple an alien invasion. And he dos all this with only a suit of powered armour created for science lab experiments, a goddamn crowbar and whatever weapons he can pick up off the ground or pry from his opponents' cold, dead fingers. But Gordon avoids the trap of becoming some kind of Gary Stu by the gameplay. Throughout the series Gordon racks up a headcount that would make the Doom marine whistle appeciatively, but if you go barelling into combat like the over-muscled space-marine you're ostensibly replacing you'll come out with the arse hanging out of your HEV suit. If you want to win at Half-Life you have to fight smart, which shapes Gordon's success- he becomes this legendary figure NOT because he's some kind of unstoppable war god, but because he's SMART. He's a survivor and he slays the Nihilanth and topples the Combine because that's what he has to do to survive.
This is the beauty of Half-Life and the reason why Gordon Freeman has become such an iconic character in gaming. He's earned his accolades. After all, didn't YOU earn them when you blew up the Citadel?