Water: While a garden hose could conveniently provide all of the water you will need, in most cases this will prove impossible. Therefore, you will probably need to bring water with you — and carry it from your parking spot to the gravesite — so plan accordingly. Five gallons (~19 liters) is easily the minimum you should bring to clean even the smallest of cemetery markers, but the more water you have available, the better. A glass or plastic carboy, like those used for home or office water coolers, typically holds about 5-6 gallons (~19-23 liters) and is convenient to transport.
Sponge: You will need at least one natural or “sea” sponge to help you wet/wipe down the grave marker during cleaning. Avoid using those cheap colored sponges because the dyes can actually transfer to stone and stain it, and resist the temptation to use one of those two-sided kitchen sponges with the “scrubby side” for the same reason.
Scraping Tools: Moss, lichen, algae and other organic growths flourish on the flat surfaces of gravestones, monuments and other markers when damp/shady conditions exist, as well as those hard-to-clean areas in/around inscriptions. Therefore, you might need a few wood or plastic scraping implements (do not use metal) of varying sizes to help remove these growths. Popsicle sticks, wood or nylon kitchen spatulas, bamboo “kabob” skewers, plastic toothed scrapers designed for cleaning cast-iron griddles, etc., can all prove handy.
Brushes: You should bring at least two brushes — a toothbrush and a larger scrub-type brush with soft bristles made from nylon or natural fibers. Like your sponges, these bristles should be uncolored so the dye doesn’t transfer to the stone and stain it. Whatever you do, do not use a brush with metal bristles, such as a BBQ/grill brush, or steel-wool pads, which can scratch the polished surface/coating of stone and metal markers.
Towels & Other Stuff: Bring plenty of old towels you don’t care about in order to dry off the cemetery marker and your hands. If you have sensitive skin, or just don’t like getting dirty, rubber gloves are nice. You’ll need at least one bucket to hold water and make it easier to wet/rinse your brushes and scraping tools, but two is better. A plastic trash bag will prove handy for hauling away dead leaves, old flower memorials, soiled towels, etc. Finally, bring a camera with you and take a photo of the marker or monument before you clean it, and then afterward. You will feel amazed at the visual differences between the two images — all due to your effort and care!