I can try to provide a few pointers but others who actually do this work could do better. One paper on the topic is:
Article Biological Aspects of Date Palm Dust Mite, Oligonychus afras...
This work does one unusual thing (rear the mites on an alternative host) and one standard procedure (the leaf-cuttings in petri dish method). I expect that the alternative host is a way around using more expensive or large host plants to rear the mites and may be the only option if the mites won't breed on young date palm, or that young palms are too expensive to use. The risk is that the mites undergo selection to their new host and thus don't reflect field populations. Sugarcane and Bermudagrass may also work as cheap alternative hosts.
The petri dish method is a typical way of conducting experiments on spider mites. It's debatable, however, that what happens on cut leaves on cotton wool in petri dishes in CT cabinets is transferrable to field conditions.
As @Owen has mentioned petri dish method is a good one. I recommend contacting @Marwa Ibrahim, Cairo University ([email protected]) who has her PhD was about the palm pests, and ran out biological experiments of date palm mite Phyllotetranychus aegyptiacus (Tenuipalpidae) on palm leaves using petri dishes methodology. Dr. Ron Ochoa, USDA, USA has been working on research project studied the date palm mite Raoilla indica (Tenuipalpidae) since the beginning of the 2000s, you can check his list of publications.
Furthermore, Ben Chaaban, S. (Tunisia) has published a good list of research studies that basically investigated the Old World Date Palm Mite (Ghobar mite). @Mohamed W. Negm, Ibaraki University, ([email protected]) had studied the Old World Date Palm mite biological control while his stay in the Alatawi lab., Riyadh, KSA.