Why the pea plant
Mendel did not pick the pea by accident. Six things about it made the experiment possible, and a worse choice would have buried the answer.
It grows quickly
A short life cycle, so several generations can be raised in a few years rather than a lifetime.
It is easy to grow
The plant needs little care and can be raised in quantity in an ordinary garden.
Plenty of seeds
Each plant yields a large number of seeds, so each cross gives enough offspring to count meaningfully.
Clearly contrasting characters
Each character comes in two sharply different forms — tall or dwarf, round or wrinkled — with nothing in between to argue about.
It self-pollinates
The flower is bisexual and closed, so left alone a plant fertilises itself and breeds true.
But it can be cross-pollinated
By removing the anthers and dusting pollen on by hand, the experimenter chooses the two parents exactly.
The last two together are the key
Self-pollination gives Mendel pure parent lines he can trust; hand cross-pollination lets him override that whenever he wants a particular cross. One plant offering both is rare, and it is the reason the pea worked where other species would not have.
What a blurry character would have done
Suppose the two forms had shaded into each other, as human skin colour does. The offspring would have formed a smooth range, no group would have had a clear boundary, and no ratio could have been counted at all. The sharpness of the pea's characters is what let the numbers speak.
The organism is part of the experiment
Mendel chose a plant that grows fast, gives many seeds, has sharply contrasting characters, and can be made to self-pollinate or cross-pollinate at will. Choosing well was half the discovery.