I do think age is a big factor. I've never even heard of rune giants or tane, but ogre magi/oni have been in the game since at least 2nd Edition. Of course, plenty of old monsters do not get traction.
The older or more widely-known a monster is, the more likely it will be seen outside of D&D/Pathfinder. A lot of monsters (dragons, goblins, etc) appear in video games such as Warcraft III/World of Warcraft, and appear in non-D&D fiction. (Shannara, for instance, has ogres and "gnomes" as villains, though gnomes appear to be goblins.)
The biggest exception I've seen to this rule are fairies. Fairies are old and well-known, but they're really hard to get right behind the DM screen, and there's so many types players (and DMs, probably) can't keep them straight.
1) Being well-known and having a simple hook. The four base races (humans, elves, dwarves, halflings) have simple hooks, and even then halflings tend to be left behind. (Humans don't really have a hook, but virtually every campaign setting puts a lot of work into differing human cultures.) Existing before D&D helps too. For many editions, dragons in D&D have made for uninteresting encounters, but just by being dragons they stick in the mind.
Other races (gnomes, half-orcs, etc) tend to have fewer hooks or hooks that are too complex (D&D gnomes suffer heavily from the latter).
To an extent, you can see this sort of thing completely outside of D&D. Daleks of Doctor Who, for instance, draw a lot of attention. They have very distinct (as a species) personality, are utterly evil, clever in the long term (but not short term), shout "EXTERMINATE!" in a creepy voice (in fact, they just have a creepy voice period) and have very distinct strengths and weaknesses (although the latter are getting phased out). Also, they were present in the second serial (so, 7th episode) of the original Doctor Who. I heard about them long before I watched the series. All those factors help them stand out compared to the mind-numbingly-large array of aliens and monsters The Doctor and his companions have had to deal with.
(One of the Doctor Who RPGs even has a very simple flowchart that describes Dalek behavior. Yes, "EXTERMINATE!" is actually one of the actions.)
As I mentioned above, age is important. It applies to being well-known. Unfortunately, lots of old monsters don't have a good hook and so aren't well-known. Gricks are an old monster, but they don't have a good hook, which contributes to them not being well-known. Mind flayers, by contrast, are also old, but they have a good hook (weird mental powers and they survive by eating brains; making them Cthulhu-oid only makes them cooler). Beholders are, well, some of the strangest-looking monsters you will ever encounter, with a wide and weird array of powers, and have a unique set of mental problems (pretty similar to Daleks, actually, only they're slightly less aggressive).
2) Homebrew monsters can't be iconic outside of a particular campaign/group of players. Did you mean which iconic monsters do you use in a campaign?
