Props are any small objects that are used by actors in a play to help them tell the story, which exclude the set design. If we were to use as few props as possible with minimal set design, there are a few objects that would be useful for telling the story of A Midsummer Night's Dream.
The most central prop would be a purple flower to use as the flower that has been enchanted by Cupid's arrow. It would be preferable to use a pansy as Oberon refers to the flower by the name "love-in-idleness," which is another name for "pansy" (II.i.171). We also know that the flower should be purple because Oberon describes it as being "purple with love's wound" (170).
For the mechanicals, two important props would be a scroll upon which their names are written and the script of their play that Quince has written. We know that the scroll is important because, when we first meet the mechanicals, Bottom asks Quince to read the list of players and their roles. Quince holds up a scroll, saying, "Here is the scroll of every man's name, which is thought fit, through all Athens, to play in our interlude" (I.ii.4-5). We can also visualize that each mechanical either has their own copy of the script Quince wrote or is sharing a copy because we see the mechanicals working through their set problems and their lines together in their rehearsal.
Other props that would be important to the mechanicals to include are a lantern, a bush, and a dog, possibly a toy dog. These are important because Starveling plays the part of the moon. To play his role, he brings on stage with him a lantern to represent the moon's light, and for unknown and consequently comical reasons, he also brings with him a bush and a dog, as we see in his lines:
All that I have to say is to tell you that the lanthorn is the moon; I, the man i'the moon; this thorn-bush, my thorn-bush; and this dog, my dog. (V.i.256-258)
Hence, we see that also having a thorn-bush and a dog as props are very important for telling the story and capturing the comedy.