"Do your own research," they say when you ask for evidence or support for whatever conspiracy theory they're advocating. "I'm not going to do your homework for you," is another one they'll sometimes throw in.
This is an evasion tactic, of course. If they had good evidence and understood it well enough to be so confident in their conclusions, they'd be able to explain it to you. But it's not just about evasion; it's also about posturing, trying to imply that they're smarter and more informed than you and their time is too important to be wasted on imparting their vast knowledge to you.
As a general rule, if you make a claim then the onus is on you to provide support for the claim, so telling someone to "do their own research" when you tell them the latest conspiracy theory is just lazy bad form. But the reason this ploy resonates is that "do your own research" is generally good advice. The trouble is that the "research" they've done usually amounts to following whatever rabbit hole the YouTube algorithm generates for them, and those rabbit holes often lead to echo chambers.
So what should it actually mean to do your own research? Broadly speaking, it means to gather evidence, evaluate its credibility and weighting, and attempt to synthesize it all into a coherent theory that allows you to make reliable inferences. The precise methods you use to gather primary evidence may vary from field to field, but in practice most of the evidence you gather will be the reports or testimony or work of other people, and so a good part of your task will be deciding how reliable these sources are.
And here's where the "do your own research" crowd go horribly wrong, because the source whose credibility it's most important to evaluate is yourself. When they exhort you to do your own research, there's often the implication that it's lazy or stupid to trust the research of other people, as if your own research is inherently better or more reliable. But your opinions and beliefs are just as likely to be wrong as anyone else's, and you should not treat them as the answer key that everyone else's answer should match to be deemed reliable.
Here's an example I've used before. A doctor recommends surgery, and describes the procedure to you. Upon realizing that she's proposing cutting into you with a knife, you reject her expertise, because even you as a non-expert know that cutting people is bad. This is the wrong way to judge the doctor's expertise, because while it's true that any expert should know that cutting people is bad, the true expert may also know other things (such as how to stitch people back up, and how to cut them so as to make stitching the back up easier) that make it not so bad.
Instead, the better approach is to say, "Wait, you're going to cut me with a knife? Won't that hurt? Won't that put me at risk of bleeding to death?" and listen -- listen -- to understand the answer. Obviously if the doctor is surprised to learn that cutting people is generally bad, they're probably not a real doctor, but a real doctor is prepared to answer this sort of question honestly and reliably. You don't necessarily need to understand every detail of what they're telling you, but if you have a good basic lay-person's grasp of the vocabulary and subject matter, you should be able to tell when they're just making up stuff and bluffing.
The trouble is, not everyone has a good basic lay-person's grasp of the vocabulary and subject matter, and even if you do, interrogating an expert to decide if they really know what they're talking about takes a lot of effort and time you don't always have. This is why we have shortcuts: credentials and certifications. If someone has a PhD from an accredited university in the subject matter, or a license to practice the profession in question, it's reasonable to assume they are in fact qualified experts. It's not an absolute guarantee, of course, but in general it's reasonable to defer to their judgment in their particular field, and checking out their credentials usually enough to qualify as having "done your own research" before you adopt the results of their research.
It's important to be clear here. You do have to trust your own research in one sense: whatever conclusion you adopt, whether it originates with you or some other source, it is inescapably the result of your decision. You're stuck with it; there's no "I was just following orders" absolution. You have a choice what conclusion to adopt, but you cannot choose not to choose; even suspending judgment is a choice (and often the right one). And more often than not, the wisest judgment is to adopt the conclusion of the experts.