You probably go to the corresponding location in the Border Ethereal
The text of Blink makes numerous references to perceiving things on the plane you originated from while you in the Ethereal Plane:
[...] an unoccupied space of your choice that you can see within 10 feet of the space you vanished from.
While on the Ethereal Plane, you can see and hear the plane you originated from, which is cast in shades of gray, and you can't see anything there more than 60 feet away. You can only affect and be affected by other creatures on the Ethereal Plane. Creatures that aren't there can't perceive you or interact with you, unless they have the ability to do so.
None of these clauses (especially the ones involving a distance on the origin plane) make any sense unless the spell sends you to the Border Ethereal in the same location as you were in your plane of origin. I can't think of any other interpretation that wouldn't turn at least one of these clauses into nonsense. Unfortunately, the spell doesn't come out and just say it, so the best we can do is infer the assumption that the spell appears to be making.
Beyond being the only consistent interpretation I can think of, arriving at the corresponding location in the Border Ethereal seems like a pretty good "default" for any travel to the Ethereal Plane by any means that doesn't specify a precise destination.
Could the DM rule otherwise?
It would technically be consistent with the spell's text if casting it sent you to a location in the Border Ethereal that was 100 miles north of your departure point. You would be able to see into the other plane at this new location, and no one at your departure point would be able to see or interact with you at all, even if they had abilities that allowed them to perceive or interact with Ethereal beings. In addition, such a ruling would have some bizarre interactions with spells that create enclosed spaces of magical force, such as Forcecage or Tiny Hut, since creations of magical force extend into the Border Ethereal, and having you appear elsewhere in the Border Ethereal has the potential to put you on the opposite side of such a barrier, and in the case of Forcecage this would trigger the Charisma saving throw for attempting to leave the cage by planar travel.
Hence, by far the simplest ruling to adjudicate is that Blink sends the caster to the Border Ethereal in the location that exactly corresponds to the location from which they departed the origin plane. Since DMs rarely want to create extra unnecessary complications for themselves, I'd expect that this is also by far the most common ruling in actual play.
What if your current plane doesn't border the Ethereal Plane?
That's a good question – so good, in fact, that it has been asked before on this very site.