the original cleric from Grognardia was of Peter Cushing as Abraham Van Helsing, the intrepid vampire hunter of Bram Stoker's Dracula. Generally I like this, because it makes much more sense as an adventuring class to have a "monster hunter" type. It's also cool to have a class with the ability to directly avoid combat, if like me you think of D&D as more of an exploration game and less of a series of carefully planned out fights.
Another thread of clerics that I really like is the Knights Templar. I wrote a bit about this in the OD&D setting series ("Clerical Strongholds") and it's a vibe I have always really wanted to play off of more. But clerics usually get bogged down in whatever fantasy excuse you have for a religion, and at level 2 (level 1 in AD&D) become the healer more than the knight.
What I'm thinking of is shifting the focus of the cleric a bit. Undead are okay, but I really like the idea of clerics as hunting demons and other "invading" extraplanar horrors. This is explored lightly in some editions, but I am thinking that it would be a more interesting main focus in a game where monsters are more fundamentally weird: from first level, clerics are able to repel or command (Lawful or Chaotic) entities from other worlds, whether they be certain types of oozes and slimes or at the top levels things like balrogs or some of the Lovecraftian horrors. (For more on this check out my series on higher planes.)
One thing I feel should flow pretty naturally from this type of cleric is that such types should be on the fringes of religion. Once you start looking at the history of religious orders in the real world, they're not always perfectly in line with the religious hierarchy. The Templars themselves were accused of being heretics at points in their history. When you have characters who look into monsters and how to fight them, there is always the chance that they will become monstrous themselves.
An archetypal cleric, then, is on the fringe of his organized religion, always a step or two away from excommunication. Not all among the clergy feel that his powers are miraculous; some suspect arcane trickery, and others even worse - that they are derived from the monsters the cleric spends too much time investigating. Commoners alternately love them as protectors and fear them as virtual heretics - sometimes both at once.
I think riding the "Templar" angle even a bit further lets us see clerics as intensely mysterious. They have a cultivated mystique and are occasionally involved in doings which they would rather not speak about. This could produce some interesting tension if the cleric's order occasionally has them go off and deliver mysterious parcels or deliver messages to powerful individuals. The idea of this "order" is backed up in the OD&D rules, which specify that clerics receive help from "above" - unless it is raining gold pieces from heaven, this could mean that the cleric is explicitly helped by their order.
To bring this back around and tie it together - this is intimately tied up with the law versus chaos conflict. Demons are intensely Chaotic, and templar-style clerics are Lawful to a fault. The presence of clerics in the world is a symbol of the dangers that threaten it, and they are the hunters of that dark night that now haunts the land. I think making some types of "lesser" demons with variable powers is an interesting way to integrate this into the cleric's life from the word go. One way to start is with the spectre, which is explicitly the nazgûl from Chainmail, and extrapolate the lesser demons downward as variant undead. It'd certainly be more interesting than the umpteenth time you've seen skeletons.