Making academic contacts (some thoughts for new researchers)
Say you've never written a paper, and may not yet have the time or readiness to solve major problems on your own; but you want to have real scientific conversations and perhaps even collaboration with a busy researcher you admire. What could you possibly talk about? What can you offer them? Below is my advice.
1) In my view, the best approach to making contacts is closely related to how I'd recommend spending your own study time as a beginning researcher: (a) start reading research papers as soon as possible, and (b) try diligently to ask interesting follow-up questions to the papers you like.
Asking questions is a central research skill. It is one you can start learning early on because, even in technically difficult fields, simple patterns of questioning often recur in many forms across papers. E.g., in the theory of computing, if there is a new model of computation being studied, it is likely to have randomized or nondeterministic versions that can be studied next. Or if there is an algorithmic problem, it might have variant problems to be solved in, say, the communication protocol or decision-tree models. And so on.
Even though it is not so difficult to produce these variations, few people do so systematically. If you keep at it, and keep trying to learn and adopt more patterns of questioning, you will hit upon interesting and solvable problems. You will also start to develop a taste about which problems are likely to be most interesting, which are too hard, and so on. (Side note---I believe that keeping a research journal with your questions greatly helps this process.)
2) Now returning to the goal of making contacts, I think that one of the best kind of emails you can send is one that contains a good research question, that's reasonably related to the person's research area. If you can do this and pique their interest, they may well enter into a dialogue that could become a full-fledged collaboration---regardless of your credentials on paper. After all, you've already brought something important to the table.
A key advantage of this approach to making contacts is that you're aiming to attract the researcher's own interest and curiosity, rather than just asking them for something. Another advantage is that there is a lot of freedom in asking and considering research questions. If you ask someone an interesting, specific question about topic X, there is no presumption that X is your only interest or that future interactions will be limited to that. And in the course of your correspondence, you might realize there is a better question to ask about X, or you might getting around to asking something about topic Y as well. That's how scientific interactions go.
So you don't have to worry about choosing the exact right question to represent yourself, as long as it is good and leads to discussion. In this respect it's less stressful than trying to introduce yourself by defining your whole research outlook. And it's certainly more promising than suggesting a collaboration based on your GPA or work experience.
Of course, asking good questions isn't easy and takes work. You should think carefully about them before sending, try your best to answer them yourself, and see if there are initial observations or partial solutions you can provide with your question to show you're serious. Maybe in the end it will be a question they can answer right away; but again, it could still lead to other questions. There is little risk in trying and it is likely to at least be a useful exercise.