I definitely don't want to lie to potential future employers, is there a particular way I should handle any questions about the gap?

Nobody will ask.

If they do: you took some time to relax after school.

I'd say the opposite, put it on your resume because at this point it's experience that makes you stand out compared to other fresh graduates. Any and all experience is important to landing a job at this stage of your career.

Say you started or finished a couple of months earlier so it looks like a 3/6 months contract and tell employers that the contract wasn't renewed because they ran out of work, which is probably half true anyway.

You shouldn't lie but this is massaging the truth in your favor ;)

I agree with putting it on your resume. This is great experience if you can be honest in your self reflection. Ie realize remote only is probably not a good idea for your first job out of school. Realize that you’ll need mentorship. Etc.

I have no problem leaving the job off the resume (“resumé” means “summary” not “comprehensive”). If asked you could say that you listed all experience that you felt was relavant. But personally I think it is a valuable thing to have listed.

However, I strongly discarded with the parents suggestions for “massaging” the resume. Those sugggestions are pure lying. There is a big difference about highlighting certain aspects of a career and stating things as fact which are not true.

You can just say you took some time off to relax after university, travelled and/or worked on some personal projects. Should be sufficient explanation, nothing strange about that.