Apparently, broken handle of the hood mechanism is a common problem for E28.
Replacement is almost straightforward.
The mechanism (part #51231868556 for US vehicles) is still available from BMW.
What is not immediately obvious, though, is that it would be a good idea to replace the nipple (part #51231872968) and the grommet (part #61138460102 or #61218050550) at the same time - there's a good chance that the nipple is broken or about to break like this, and the rubber grommet has deteriorated enough to warrant replacement.
In my case, part #51231881741 did not need to be replaced. Honestly, even the main cable (part #51231868559) was just fine, it was the left and right catch (#51231876997 and #51231876998) that were bogged down with dirt beyond reason, but alas, you can't buy just the handle.
Be careful removing lower instrument panel cover - there are two connectors on it. Careful with the screws when you put it back - some screws go into plastic, which is most probably already cracked.
