Letter of Recommendation Without Contact from the Student

I would do the "third thing" of nothing. You're not under any obligation to write letters of recommendation, and to me, the student's signing you up to do this is similar in principle to your department head signing you up for an extra committee assignment, or someone signing you up to volunteer for a public lecture, without discussing it with you first. I write this as someone who puts considerable time and effort into recommendation letters -- two that I care considerably about in the past 24 hours, for example.

Of course, it may well be the case that the student has no idea that writing a recommendation actually takes work, and it would be nice to contact him/her, but you've already tried the simple routes to this. I would devote my time to other tasks.

I would just be honest. You teach many students and this particular student you taught a long time ago. Their grades were X and that puts them in the top Y percentile of the year, etc etc. No record of academic misconduct. Seems like one of the better students to you.

What I wouldn't do is try to fake all the usual character stuff by having a chat with the student over Facebook. Your intentions are noble, but I don't think the outcome will live up to those intentions.

That's happened to me in the past, and from a student that I would not have written a letter for. In that case, because I could not endorse the student, I contacted the student and told him that I would not be responding to the request in any form, even just to tell the requester that I would not be providing a letter.

In this case, the student seems average, and had the student complied with your process, you might have agreed, so the situation is different. That said, putting down your name as a recommendation without your consent or even contact (... though you might check your junk mail folders) does not convey a level of maturity that would make me comfortable endorsing the student.

My own action, given that you've already done your due diligence in trying to contact the student, would probably be to take no action, and completely ignore the request -- at least until you've been contacted by the student. If contacted by the student, I'd probably say that I was unable to write a letter.