I kind of feel like I should say something about Dawson as I'm constantly bringing up that this is a franchise that includes the Expos history. Thing is - I didn't start following the franchise till 15+ years after Dawson left, my consistently failing memory only vaguely recalls "Expo Dawson" and most importantly, I wouldn't have voted him in if I had a vote. So what's a boy to do?
Dawson was a borderline case mainly because he didn't get on base and we're in a time where we realize that it's very important to get on base. But while that hurt him offensively, it didn't make him a bad offensive player. See a .328 OBP is pretty damn bad, but there's a distinct difference in being a guy with a .328 OBP who hits .260 with moderate power and a guy with a .328 OBP who hits .280 with fantastic power. The former is a guy who is probably "secretly" hurting the team, the latter is a guy who is very good rather than great. 16 straight years with an OPS+ of 100 or better, 13 of those were above 110, 7 above 130. He played over 138 games 13 of 15 non-strike years. Dawson was very good for a very long time.
Add to this that through the first third of his career he was a legit great defender and stolen base threat, and you have yourself someone who is good enough to be a hall of famer. Of course this was the case a decade ago when he first appeared on the ballot, why does he only get elected now?
Getting into the Hall of Fame amounts to convincing 2/3rds of the voters that you were great. (once you get to 2/3rds there starts to be a "I don't want to be the jerk who keeps him out" effect on the remaining voters. Peer pressure takes over and it doesn't matter if they feel you fall just short.) How do you convince a voter you're great?
Be undeniably great - be so great that any "but" in the conversation feels like an attack rather than an argument. Hit all the traditional numbers, hit all the sabermetric numbers, win a world series if you can help it. Ruth, Mays, Aaron, coming up Randy Johnson and Maddux.
Be undeniably great for your position - if you can be undeniably great with qualifications, you should still get in. For non OFs and first basemen you can get by on being awesome in comparison to others that played the same position you did. You may lose that "first ballot" vote, but you'll get in. Roberto Alomar, Carlton Fisk.
Be undeniably great at one thing - be the best fielder ever (Ozzie) or the best strikeout guy (Ryan), steal a ton of bases (Brock). Usually this skill will keep you in baseball long enough to hit other reasons for being elected - but this helps push a player into first ballot territory.
Be really good for a really long time and hit those magic numbers - 500 HRs, 3000 hits, 300 wins. It's tough for voters to deny these things. You are great by the sheer force of numbers. (and good - I hate the animosity toward "compilers". Most players play until they just can't do it anymore. If you happen to be major league worthy longer than someone else - that should matter.)
Usually it's some combination of the above that gets most players in. But what about the borderline cases? The ones, like Dawson, that don't fit into the above categories. They need some extra help.
Be great at the right time. This is what got Jim Rice in. The common view is that he was the "most-feared" hitter in baseball for 5-7 years, but really he wasn't even the best hitter in the game for his admittedly awesome '77-'79 stretch. However. in 1978, when all eyes were on the Sox and Yanks yet again, he WAS the best offensive player in the league. So a reputation was born. This would be cemented when later he had another very good season in '86 when the Red Sox would get one out away from ending that horrific/awesome drought and everyone was focused on them again. Those memories stick. It may be being dominant when your team is also dominant (Ecksersley), it may be a great playoffs (Puckett). If you can get in people's heads that you are great when they see you - they won't care as much how you were when they didn't see you.
Be the best thing on the ballot. It's hard for some voters to believe that there were times in baseball where no one was great. So if you hit a lull in actual great players, and you are merely very good, you start to look great in comparison. Out of the top 200 OPS+ seasons (so these are adjusted for time period) only 14 fall between 1972 and 1991. If these seasons were distributed evenly we'd expect around 37 in those 20 years. Even worse, there were only 5 between 1974 and 1984 (20 expected). Some of these great years are from obvious non Hall-of-Famers: Bobby Murcer, Fred Lynn, Kevin Mitchell, Jack Clark. Objectively there just wasn't alot of great offense in these years even taking into account the era. Yet it's hard for a lot of voters to believe that. So you get guys voted in like Rice, like Tony Perez, like Andre Dawson.
Be remembered for something that was great. Maybe you weren't great but something you are linked to is. You get points for "inventing/popularizing" pitches (Sutter, Candy Cummings) for dramatic singluar moments (Fisk waving that home run), usually this alone won't get you in, but it could be the final boost to get you in.
Be better in comparison to someone in the Hall of Fame. This is another "great by comparison". Sucks, but it's true. When someone gets in because of one of the reasons above and you can clearly show you were better, well you must be great too? This is what's helping Blyleven, and what helped get Dawson in. If you lower the bar, then certain borderline candidates drift right over the threshold.
These later reasons can't do it on their own. It still needs to be agreed that you were at least very good for a decade or so. But with these reasons you can get to that point where some people start thinking that you were great - and if you can get 2/3rd agreeing to that - that will get you into the Hall.