I'm looking at the rod and it's not 1/4" diameter so the thread won't be 1/4". I can't get a caliper down there to measure properly but I suspect its a 3/16" rod. If so, I would think the thread would be 10-32 or 10-24. Go to Home Depot and find a nut that fits. Once you know the size you can do what other people have suggested, filling it in with JB weld and then re-drill/tap. The other option would be to put JB in there and just press it back on. If you do this you'll have to be sure to wire the button in the pressed in position for the 24 hours or so to cure. If you don't the spring of course will push it back off.

Hope this helps. I do have a replacement if you need one.

Ron

fill it with JB Weld.  It's great stuff and you can drill and tap it.

Dig a little more digging and yes I have a few spares. Email me.

ron

Sorry - only have one and it's on the Moke!

updated pictures of the brake in the 1967 Moke.. white button came out.. cannot get back in. will try again but if a spare brake like this one is available I would be interested

You can send me an e-mail to coordinate a get together over there.  It may take a few days to make it work.  In the meantime I can sell you some wheel chocks, but I bet they would be more expensive than the part that I will give you for free if we have it.

I might have spoken too soon. I did get one in a bunch of parts so I thought I had an extra. In checking my pictures of an upcoming Cooper project I don't see that it came with one. I'll keep looking though, there's still a chance...

i will take some more pictures.. maybe it just looks longer because of the Bob Barker, Price is Right type microphone for a shifter in the moke.. and if we cannot fix what would the price be for that Tagus.

thank you everyone for your help.  I really miss Robin Jones.. if Robin was here I could drive to his house on the way to my mom's and he would take a look at it, walk into his garage and come back with a fix.. feeling sad, not from the brake but from not being able to see Robin and hang out and talk Mokes.

I can probably help out with a replacement assembly if needed. Shipping would be from western Canada though...

Ron

Sorry I did not notice that.  That's the one with the short cable then the wheel to wheel cable on all minis after May 1976,

 Jim:
     Please post a picture of the complete handbrake - is it a two cable or single cable.  You say big long handle but is it the same handbrake as used in all the other mini models?  As I recall the early Minis (Mk1 & Mk2) used the white button you are looking for.  Some of the variants Austin America, MG1100, etc had a chrome button.  Some of the rods were threaded, others not.  Some made modifications to their Minis by changing them to the fly-off handbrake and Paddy Hopkirk actually offered a conversion kit.  

If you do not have the repair capabilities Jim just get a used handle they are easy to install.
For the time being just leave it in gear when parked.

thanks for all your tips.

The hardware store sells long nuts used to join sections of threaded rod that could be used to verify the thread size and pitch, then just heli coil in new threads, seems it would be nice to keep the original button.

The knob is threaded. I recall it being 1/4x28 but I could be off. The idea of using epoxy is the best idea. Fill it in, let it try and drill a hole smaller than what you need. Then use a tap that is the correct size and pitch. You can even drill out the rivets if you want to access the rod itself to determine size and pitch. Go back with machine head screws and nuts or use industrial rivets if you have access to them. 

that link says 1976 onwards

ebay.uk
https://www.ebay.co.uk/itm/austin-rover-classic-mini-handbrake-lever-with-narrow-grip-76-onwards-/332476349814

jim.. new ?? I doubt it. but might try.. www.m-parts.co.uk  www.runamoke.co.uk or even leo jack at www.mokepanels.com   

is a replacement available ?  i will not be able to perform those steps on the fix but i can check with my mechanic that is real good if a replacement is not available.  thank you

JimOhio.. the last braking brakes and shifters I bought used from Dion..    these was when I changed both out of the 2nd  Chokes.. later bc