Here are a couple of thoughts that might help curb resource depletion while still encouraging multiple sector claims:
1) Comets and Meteors
I don't think it would take up too much of a server's resources to spawn an asteroid 'ship' that flies through a sector. Like pirates, this only needs to happen around where players are currently playing - not be a persistent entity. They spawn, fly a path through a sector, then despawn if they are far enough away from players. Players can mine them as they fly, or use kenetic weapons to catch them. They have a spiral path towards a star or worm hole (where they despawn outside player's vision).
The more players logged on then the more sectors that the server must keep loaded. The more sectors currently loaded, the higher the likelihood of a comet/meteor event. The more space a faction owns, the more likely they are to have a member find a comet/meteor.
2) Gravity Well Events
This is more fiction than science, but it will work for a game. In the real universe stars explode and reform - space is a vacuum, but it still has dust in it that (over time) condenses into a new stellar formation. So rather than asteroids just 'respawning,' they are being formed naturally by gravity. In essence, they grow over time (just like reality, but over the course of hours/days rather than millennia).
A sector of space has a chance of spawning a 'gravity well' which shows up on scanners and plays havoc with thrusters (like a weak black hole that kills instead of teleporting). This area grows hazy/foggy as it collects space dust. Eventually the dust settles and an asteroid (or cluster) is there. The event becomes something to fight over/defend since it takes a long time to 'finish'. It is rare that it starts, and takes a long time to finish.
The more sectors a faction owns, the more likely they are to have an event naturally occur in their area or near their scanners.
3) Gravity Well Modules
If #2 seems incredulous, then perhaps we could instead say that the event is man made. We have a Gravity Well Module, which is shot into space like a decentigrator. When it is detonated (by laser or missile) it makes a couple of checks:
A) Only one gravity well active per sector
B) No entities within X-meters
C) The sector is owned by the correct faction (determined by faction module attached to gravity well module)
If everything works out, it creates the hazy/foggy space-dust collecting event I described above. To prevent it from being abused the gravity it uses to collect dust shouldn't affect other entities. Over the next few hours (or whatever time is appropriate) it collects space dust and turns into an asteroid. I see this as a faction moves to an area, launches the well and activates it, then leaves and comes back the next day to see what formed.
The price of the module used to do this is the control over how often it will be used. It is faster than flying 40k km, but also much more expensive. If we limit them to one active per sector, then a faction can have more asteroid 'farms' if it owns more territory. We could also say that the color of the nearest star determines the probability of certain ores developing - if you want a particular ore you should find a particular colored star.
EDIT:
4) Order an asteroid from an NPC. You always pay a set price, but you never know what you will get. Sometimes it's huge and ore-stocked, other times it's small and basically useless. It's a gamble, but it is always interesting to watch the Trade Guild use their push beams to fly an asteroid towards a store. You also have to be careful, because the Trade Guild just does delivery. If someone else ambushes them, or starts mining the asteroid with you, well that's not their problem.
You also have to wait for the ships to arrive. It might not be instantaneous.
