I don't think your request is odd. I do telecom for a health network and we have a customer service initiative called "Legendary Impressions" - we want to go above and beyond the expectations of our customers so they talk about how great we are - it does have a positive impact.
We have special dialing procedures for patient rooms for long distance calls, and I have had custom made die-cut labels produced that explain to the patients how they need to dial LD calls and 800 calls. We place them on the handset of the patient phones, but they are sized about the size of a standard address label and they fit equally well on a standard handset.
We also have other special labels made for "Rapid Response" that are much smaller and will go almost anywhere on a phone. I'm not exactly sure where they are getting those from.
It would not be inconceivable to have a small sticker on the phone in a spot near the display if the phone has one (because they will look there) that says:
Answer with a smile
In your Voice
The place I use for the larger die-cut labels is McLoone Metal Graphics - they are near LaCrosse WI. They make all sorts of labels and tags, etc. The ones we have appear to be vinyl with out info silkscreened or web printed on them and then there is a clearcoat over the top. They are very durable and they don't make the phone skanky when you take them off unless you yank on them and rip them really hard - then they will leave some adhesive behind that cleans up with an alcohol wipe. I buy my labels in lots of 500 and they last a long time. I typically replace about 60 patient phones a year that get dropped on the floor and broken, just wear out, or worse. For $12.50 each if someone barfs on a phone it's going right in the trash chute!!