This is a production made Nostromo Cap prototype.

I say "prototype" because I won't pretend it was screen used as some of its details show it definitely wasn't. The Nostromo patch and "scrambled eggs" simply don't match the ones seen on the cap worn by Brett in the film… but after all that was also the case of both Nostromo caps auctioned these last years by the Prop Store of London and Profiles in History. They didn't match the screen used cap neither…

The truth is (from what I know of course) that the screen used Nostromo cap worn by Brett (Harry Dean Stanton) never surfaced on the market.

But let's go back to the cap presented here : It's a black trucker style cap (the cap seen on screen was dark blue) with no tags inside. It has a genuine production made USCSS Nostromo patch glued to its front.
The patch closely matches another patch I have in my collection that was directly acquired from an A L I E N production vet who had kept it since 1979. Both patches are rather close in size to the ones made by The Thinking Cap Co. in 1980 but their quality and details are imho closer to John Mollo's original design, thus indicating they were production made.
The Nostromo patch on that prototype cap has a dark background, when the screen used cap had a patch with a light background.
The yellow scrambled eggs embroidered on the brim are very different from the ones seen on screen, which by the way were probably made of golden thread.

My best guess would be that this cap was one of many attempts made by the Costume Department to define Brett's cap, like the aforementioned ones sold by PSOL and PIH.
To do so a black cap bearing yellow scrambled eggs was used, on which a Nostromo patch was simply glued in order to give it a "US Navy of the future" vibe… "Captain caps" bearing ships names and scrambled eggs were already worn in the US Navy at the time A L I E N was in production. I think this is the main idea behind this piece of costume : evoke a casual cap worn by crews operating for a huge commercial star fleet.
The cap has also been lightly weathered, giving it an interesting semi worn-out look...

It was acquired from someone who was a kid at the time A L I E N was in production and whose father was working at Shepperton. The cap was given to him by his father just after production wrapped. It came directly from the Costume Department and was about to be dumped with other remains of the film when it was rescued.

The cap is in very good condition, considering its age.