No, there’s no Standby tickets as such. 
 

You might find something on Eurostar Snap, where you buy a ticket but you aren’t informed of the departure time until the day before travel

https://snap.eurostar.com/uk-en

It’s also worth knowing if you’re travelling on a pass, that passholder fares often become available a few days before travel, even if there was no availability weeks or months in advance.

What is it you're trying to achieve?

I am a pass holder, I am traveling to get to London from Berlin on the 22nd - it seems like all the trains are good to good in order to get to Brussels on that date, but then there are no more tickets Brussels to London (Eurostar) for 2nd class pass holders on the evening of the 22nd, I booked an early Eurostar on the 23rd, so my current plan is to spend the night in Brussels. 

But I am wondering if there is a possibility if I am already at the station on the evening of the 22nd, that I might be able to hop on the evening Eurostar to London -- if there is a space that becomes available?

The trains aren't full yet, and sometimes Eurostar open up extra pass holder seats from/to London at short notice. The best is to keep checking availability, e.g. via B-Europe, and change your Eurostar reservation as soon as you see something. Changing the reservation will cost €15 per person and can be done on the Eurostar website, with "Manage your booking”.

Also it doesn't hurt to ask at the check-in. They might allow a change even if there are no more pass holder seats.

That’s right after the Easter weekend, so trains will be busy. I still think there’s a decent chance that availability will appear for the 22nd.

Keep checking for passholder availability. As ​@rvdborgt says, you can make the change on the Eurostar website via the Manage Booking section.

It certainly doesn’t hurt to ask at the station if they can do it for free, but I’ve never heard of anyone being successful that way.

I don't expect Eurostar to change the booking for free, but they might just do it for the normal change fee, even if there are no pass holder seats anymore, but still empty seats.

Thanks for all the great advice!

• I purchased my ticket for the morning of 23rd through the B-Europe site, but got the confirmation through SNCB International, so not sure I can Manage my Booking via Eurostar. ​​I will try though!
• I will keep looking for seat availability as the trip approaches, and if nothing comes through, I will just try to plead with the Eurostar representatives at the station on the evening of the 22nd! There are two trains departing after I arrive in Brussels that night, so I am hoping at least one will have space! 

Yes, it will work in Manage Booking. You just need your surname and the 6-digit PNR that you got from B Europe.