Welding could be done, but it is stainless and thin too, so I'd say a job for experts only. Silver-soldering a patch on top would be a safer option IMO, and may be what I'd try, but it needs special flux and a decent blow-torch (and could still bow a little). Glue like J-B Weld may work, but I'd be suspicious of failure with the base being directly heated by the hob; claims are made for its strength, its heat resistance and its ability to stick stainless but my experience is that you don't necessarily get all of these at the same time. Similar for waterweld; it is good where it can be moulded around things and 'may' work here, but it doesn't adhere as strongly as J-B weld so personally i'd not trust it as a permanent fix on a flat, heated base with gallons of boiling wort.
If you wanted to DIY it cheaply, without any special tools and don't mind a lump inside then you could use a machine screw to fill the hole. Sadly, this is much more complicated by needing the base to be free of lumps but a countersunk screw could still be used. If the pot is thin, then rather than drilling the countersink recess (which would leave virtually nothing) you can dimple the pot to form it, then place a countersunk bit of metal (or washer) on the inside. As a vaguely similar example, this was something i did to fix a lump of stainless to the side of my pot with M5 screws, but it sounds like you could manage with much smaller so it would be less bulky - say M2 or M3:

The dimple was made by pulling a slightly larger diameter csk screw into the hole - braced by a stack of washers on the other side, with holes just big enough to accommodate the reverse of the dimple:

FWIW this (centre) was the lump of metal fitted over the reverse of the dimples - the countersinks in it were just big enough to clear the convex side of the dimples and I threaded the holes:

But for your purposes some washers or softer (more easily drilled) non-rusting material could probably be used with a nut on top. I used J-B weld as a sealant in the join, but probably fernox LS-X or food grade silicone would work also.