So I've recently graduated with a STEM degree, but my advisor still wants us to make a submission to a conference.
Right now I am unemployed and I have no source of income.
What is the best way going about telling my advisor that I would like to be paid?
I've been in similar situations, both from your perspective and from the other side, eg. collaborating with people no longer employed in my research group.
The common mindset of people in academia is that they are happy to get publications. Depending on your field, it is also not unheard of for people to pay for publications. So I'm pretty sure that the way your previous supervisor sees this is that they are offering you an opportunity to get a publication. You didn't state it in the question, but usually when people ask previous collaborators for such things, it is implied that the previous collaborator will have to do minimal work. So in this sense, such an offer is indeed very generous from an academic point of view.
Of course it is up to you how you view the situation. If you think that you would not benefit from the publication personally, then it's completely fine to say no. However I don't think you can expect to be paid, I've never heard of such an arrangement.
It really depends most on Roland's question of what is your career plan. If you don't want to stay in academia, there might be no reason to do this, and you should simply explain. If you want to stay in academia, doing this will likely make it easier for you to find a job, so you should try to find a way to do this that you are happy with.
I would consider saying something along the lines of
"I don't think I will have time to work on this conference submission because I am currently very hard at work looking for jobs."
The idea is to not accuse them of trying to exploit you for free work (as Szgoger suggests, there is a good chance they thought they were helping you out by offering you this opportunity, and might be offended by such an accusation) but still let them know about your job situation and connect it to the paper. This assumes that, if they have some opportunity to, e.g. take you on as a temporary postdoc, as Bryan Krause suggests, you would want to take it and then work on the paper.
This is an exemplary case of academia asking "innocently" for some free work.
Please keep in mind that by asking money, or at least refund of the living expenses you will have, you will be looked like a jerk by the median academic person ("what? this person wants to be paid to do research? how they dare.") while in fact it's the person proposing free work that's the jerk.
Yes, if you do that work you will obtain visbility and it will help to build your resume.
If you want to stay in good terms, ask if they will pay you the travel and accomodation to the conference (hoping they will hold true if they say "yes"). This is not payment for the work you do now, but it is at least a tangible and concrete form of future compensation for the work performed now.
