It's acceptable to put anything that you think is relevant on your resume.
You want to include details that paint you in a good light, and show that you are a potential employee that the interviewing manager should hire.
I've been interviewing and hiring for many years and have never seen this sort of thing on a resume. Frankly, I'm not sure I'd lend much credence to an offer that wasn't accepted. It would be hard to check, and would raise doubts if it was a formal offer, or just one of those typical "look us up when you are ready for a full-time position" offers that most interns get. Perhaps it was an offer at such a minimal salary that it wasn't worth much.
The most useful resume items are those where you recently worked at a job which demonstrated skills similar to the job you are applying for. Rejected offers don't really demonstrate much.
I suspect your best bet is to simply mention just the internship on your resume and cover letter, and go into the fact that you got an offer of employment from them (and subsequently rejected it) only if the internship topic comes up in your interview.
But if you are new to the job force, you may not have anything better than a rejected offer to put on your resume. In that case, something is usually better than nothing.