Depending on how your survey is setup (closed survey, open survey), there are various options.
To track responses, I would always recommend token-based surveys (closed surveys). It creates individual links and let's you track who has answered and who hasn't.
Now your case is special, because you want to track not only the students (who will receive the link in the first place, but also you want to track the 3 people they will "recruit").
I personally would provide each student with 3 individual links (3 token links).
The other option is creating 1 link per student, but increase the uses left to 3 or more (if you leave it at 1, once the link has been used once, it can't be used again).
If it is an open survey you would need to add a variable per student to each link, which at the end is basically the same as a token based survey, just with less control and more manual work. So I would go down the token route.
We do similar things in market research when we send interviews out in the field.
I usually create token based surveys for this, because if something goes wrong in the field (e.g. the internet stops working, the respondent asks to come back later, etc.), the interview can go back to a specific interview later and continue where they have left off.
As you can upload your own token table to the system, you can also create your own "tokens". I usually create a code for each "interviewer" (which corresponds with your "students") and then each interviewer gets a number of tokens.
E.g. I give letters to the interviewers A, B, C, D, and then add numbers to that, depending on the total sample (and of course always a buffer).
E.g.
A001
A002
...
A999
B001
B002
...
B999
...
You get the idea. In your case you could use a student-id or if there is no such thing, you could just create IDs that you give each student.
And then create 3 (or more) tokens with this ID, similar to what I did above.
In this case, you would not send the invitations via email (you probably could, but each student would receive 3+ emails with individual links).
Not sure how you will distribute the links to the students. This can also impact the way you want to setup your survey. So if you give us a little bit more detail on what you want/can do in terms of distributing the survey, we might be able to give more details.
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Note: I answer at this forum in my spare time, I'm not a LimeSurvey GmbH employee.