Blindsight is an amazing phenomenon with its implications for intelligence without consciousness and a great book by the same name.
Wow, go humans. But I guess this is nothing in comparison with what the commercial farming industry gets up to :(
It wouldn’t have to be powered on for the whole journey (to preserve power) as long as it pings back it’s location from time to time.
And since it’s location is known it should be possible to recover every unit so as not to pollute the oceans with them.
Shouldn’t cost more than $200 per unit, I’d even tend to say that it’s doable for around $100.
This would have the benefit that you can plot the exact course each unit has taken (in real time). And for where there’s no mobile connectivity just save the location locally and push it once connection is regained.
As others note keeping the electronics alive is going to be hard enough, but the "ping back" is probably the hardest bit.
There is no such thing as mobile connectivity in the middle of the ocean, because there are no cell towers, so your only bet is to use satellite hookups.
Satellite connectivity for things like this is pretty good today, with providers like Iridium providing low bandwidth ping services like their Iridium Short Burst Data Service.
You would use something like the Iridium 9603 which idles at 34mA and uses 0.8W to send a message.
That is to say it's all definitely doable, but probably not that simple and much much easier to do today then even 10 years ago.
It would be an improvement over having to find them manually, but if you've gone to the effort to engineer an electronics package capable of connecting to a cell tower after years at sea you can probably get it to ping a satellite regularly "for free".
In the early days of remote sensing, it was thought that these sondes, which are neutrally buoyant and can stay aloft for months, could provide enough data to constrain weather models, especially in the southern hemisphere. And then satellite data with all its calibration problems would not be needed.
It turned out that the sondes tended to drift into certain regions and cluster there, leaving large areas without data. Of course, sondes still play a key role (as do the ocean drifters), but the strategy of just using natural drift to gather data everywhere didn't work.
Not to say there haven’t been massively successful drifter studies conducted by oceanographers, though the per unit cost is well more than $200. One of the largest fleets is the Argo float system, with nearly 4000 floats in operation in the world ocean.
So - using disasters to discover new things, for science!
My favorite example of that is still the discovery of radiotrophic fungi in Chernobyl.
It's just so cool to find out about species that feed on radiation. Too bad it takes a local-nuclear-apocalypse level of disaster for that to have happened.
Edit: Sadly they only allow recent news.
Models of surface wind and currents were used to both predict, and trace back the possible origins of flotsam and debris from the crash that washed ashore around the rim of the Indian Ocean in the following months and years.
As a side note, the Friendly Floatees event occurred in the Pacific, with drift around the Pacific, Arctic, and far north Atlantic oceans. MH370 was lost over the Indian Ocean.
 Part of my sources include personal knowledge, as I lived in Canberra and knew employees at the Australian Transport Safety Bureau during the early months of the search.