Who is your audience and what is the context?
It would depend on where this notice or message is placed or received. If you can put one such notice/manual/flyer on each copy room or copier, the singular might suffice.
Remind each employee to write their name on the schedule on the lid of
It would also depend on who the audience is. If it's an email/mass notice, and you're addressing multiple shift managers or other such responsible persons, then I would word it such that plurals could be formed as needed. E.g.:
Remind each employee to write his or her own name on the particular
schedule on the lids of each of the appropriate photocopiers.
Again, lacking context it's difficult to determine exactly what this sentence is attempting to instruct.