According to the four guys at rpg.net that are still super mad that D&D 4e was replaced by D&D 5e, the thing that makes monsters interesting are the unique mechanics that express a monster's flavor. Kobolds are interesting, in this view, because they shift all over the battlefield--emphasizing what sneaky little bastards they are. Gnolls are interesting because they're pack predators and they get a mechanical bonus when fighting next to each other.
I can see how someone might see mechanics as the interesting bit about monsters in a game like 4e that uses a skirmish-level board game as its combat system. The mechanical differentiation is obvious because you see it in action as the miniatures or pogs or whatever are moved around the battle-map of one-inch squares. Certainly, it solves the "bag of hit points" problem: in practice, a bugbear isn't just a goblin with more stamina--it's got something that sets it apart. (5e seems to do this as well, at least with humanoid monsters, but I never see that mentioned.)
Also, 4e's mechanics had to do the work of making monsters interesting because whoever was writing the fluff kept committing hate crimes against flavor text like this infamous table of "Bear Lore":
|Note that 4e actually made you roll to learn this stuff. Why?|
I'm less convinced that in D&D games that aren't 4e mechanics can and should carry the burden of what makes a monster interesting. This is especially true for games that don't rely on battle-maps and miniatures; since the mechanics of pushes, pulls, swaps, slides, shifts, marks, blasts, and bursts are not defined in a tactile way in "theater of the mind" combat, those mechanics simply hold less weight when you're relying on shared imagination instead of object-defined positioning.
Of course, the notion of "shared imagination" always already suggests what makes monsters interesting in "theater of the mind" games: the interesting bit is how the monsters are described. After you've played in your tenth D&D campaign, goblins probably aren't that interesting anymore. But if you were to describe those goblins (without actually naming them goblins) as "diminutive, wizened, man-like fey, each wearing a cloth cap that appears to be dipped in blood" they suddenly become much more interesting than their stock description in your Monster Manual allows for.
Even a monster with a ton of mechanical options (such as the beholder) is only interesting until you've seen what it can do and it becomes familiar. Changing up the description ("re-skinning") can still breathe new life into what has become rote. To that end, I'm going to be posting a series of regular monsters from the 5e Monster Manual that have been re-flavored for use in my Krevborna setting to illustrate how new description and changing the script adds interest to the same-old-same-old.
My method for Krevborna is to mine myth and folklore because that fits this particular setting, but you can go further afield in your own games. Get inspired by Clark Ashton Smith, comb DeviantArt for sci-fi monstrosities to be inspired by, roll on random tables if you must; re-skinning gets mileage out of the books you already have (no need to back the latest monster book coming down the pike on Kickstarter when you can DIY) and saves you the time spent crafting your own monster stats by hand (good luck doing that in 5e, by the way).