Wire mesh flooring often gets incorporated into rabbit run design. Unfortunately, rabbits don't have padding on their feet and if forced to sit continually on wire will develop sores on their hocks and feet.
If the run has wire on the floor there should be an area made of solid material where the rabbit can sit comfortably. Wire has the advantage of allowing moisture and waste to fall out of the run easily.
Wire can simply be brushed off when cleaning.
Plastic Slat Flooring
Commercially made runs or cages have a flooring option of plastic slats. The slats are wider than the wire floor and better for rabbits' feet.
Exercise pens designed for dogs with panels that can be joined together are often used for outdoor rabbit runs. These can work well as long as they have a roof and are secure from predators.
Using the lawn as the flooring has some problems. After a short time the rabbits will have eaten or trampled all the grass, leaving bare soil.
The soil often becomes damp and muddy. In addition, unless you plan to supervise very closely, predators could burrow in or the rabbits could burrow out.
Homemade runs often utilize wood in the construction. Wood for flooring can work, but seal it for good sanitation.
If it is not sealed, moisture will seep into it, making it impossible to keep clean. Rabbits like to chew and wood floors could get chewed on.
Pressure-treated wood or deck wood should never be used in rabbit run construction because it's treated with poisonous chemicals.
Incorporate a marble tile into a run as a cool spot for a rabbit to lie if you live in a warm climate. Linoleum or other household flooring can serve as flooring in runs, but should ideally not have many seams and needs to be firmly attached to the sub surface.
This prevents moisture from seeping underneath or rabbits from finding purchase to chew. A cage with a metal tray in the bottom can work if covered with straw, but keep it very clean so the rabbit does not sit in urine.