I managed to encounter one.
First: this is something that only recently started happening and may stop happening. It appears to be a scripted event and it's likely part of some kind of ongoing plot. (in other words, if you read this months or years later, these encounters may be unavailable or completely different)
Second: it's not necessarily a Thargoid. It certainly does appear alien, but it could be different aliens or even something from humans.
There is definitely randomness involved.
An Unknown Artifact (UA) is not necessary, but it appears it may increase your odds. The majority of commanders that had an encounter had, at least, interacted with a UA in the past. So if you have never even interacted with a UA before, it's probably a good idea to pick one up. Unknown Probes may also help.
So far, all of the events have occurred in the same region of space: Pleiades, Aries Dark Region and various things around there.
The recipe I followed that worked:
- Go to Maia and visit Professor Palin's lab (requires Horizons) and purchase a Corrosion Resistant Cargo Rack. (optional, but avoids damaging ship when carrying UA)
- Go to one of the known Unknown Artifact locations and pick one up (try HIP 17862 6C A coordinates 30.32/-98.6 with SRV). (optional, but seems to increase odds)
- Plot a route to some random Pleiades or Aries Dark Region system. It's unclear if economical route (more jumps) or fastest route (bigger jumps) is more helpful.
- Go there.
- Plot a route to the furthest Pleaides or Aries Dark Region system you can easily find.
- Repeatedly go all over that area, back and forth.
(If you don't have Palin access, you can skip the Corrision Resistant Cargo Racks, but if you put a UA in a regular cargo rack your ship will take damage over time.)
https://www.reddit.com/r/EliteDangerous/comments/5m9t09/thargoids_first_contact_videos_theories/ lists some systems encounters were recorded in; you can just pick your destinations from the list (a couple seem to have typos in the name, though).
When it finally happens, the jump looks a bit different and then you get a warning about a jump problem. (give you a chance to prepare yourself and fire up the recording software)
From that point, it's very scripted and the only thing you can do is look around your cockpit. At the very end, your ship powers on and you can try to interact with it for a few seconds before it jumps away.
The only extra equipment that seems at all relevant is a wake scanner. You can scan the ship with your regular scanners, get a data material and see its outfitting (some normal stuff, 1 pulse laser and some empty slots). When it jumps away, you can scan its wake and get another data material. Discovery scanners don't do anything; warrant scanners don't find anything; cargo scanners don't find anything; and nobody has yet made a dent in its shields with any kind of weapon. Some people have tried to chase it into the wormhole/wake/whatever, but I haven't heard of anybody succeeding at that.
I'd take an Advanced Discovery Scanner along, though, since you might as well scan all the systems you're jumping in and get some exploration data.