One of the many meanings of the English noun way is, “a thoroughfare used or designed for traveling or transportation from place to place.”
Roman legions travelled along the Appian Way.
Shakespeare’s Autolycus sang,
Jog on, jog on, the foot-path way,
And merrily hent the stile-a:
A merry heart goes all the day,
Your sad tires in a mile-a. —The Winter’s Tale, IV.3.
Even now we drive along highways and perhaps hike along byways. The land that runs along these “ways” is called the wayside.
The expression “to fall by the wayside” probably came into English from the parable of the sower:
A sower went out to sow his seed: and as he sowed, some fell by the way side; and it was trodden down, and the fowls of the air devoured it. –Luke 8:5, KJV.
The idea is that anyone or anything that “falls by the wayside” has failed to accomplish its purpose.
Nowadays, speakers who don’t seem to know the word wayside talk about “falling by the waste side.” Like those who still say wayside, they are referring to the area along a road or highway, but–doubtless because of all the litter that accumulates there–the eggcorn “waste side” makes sense to them. The expression has made its appearance in pop lyrics:
When you got big dreams, keep your eyes on the prize
Don’t fall to the waste side, reach for the sky
Don’t bother wasting my time, you’re falling down by the
Falling down by the waste side
It’s not just songwriters who are mixed up:
Senator [Royce] West added that, “It was not his intent or those members who have coauthored SB 1419 and those who lent their efforts to the creation of this bill, to attempt to maintain or increase the pool of eligible applicants for admissions, admit them, and then have them fall by the waste side.” –From the Office of State Senator Royce West, District 23 (Texas).
Note: Carpenters aptly talk about the “waste side” of a saw blade. It’s the side away from a section of wood being cut.