Permanent Bracelet At Home
Possible? Probably

If you have an a soldering iron and some fine wire, you can make a semi permanent bracelet - I say semi permanent rather than permanent because the kind of soldering you do on electronics isn't really designed for mechanical strength, and unless you buy some form of welder (the article you linked mentions arc welders) I'd expect the soldered joint to break before too long.

Advisable? No

You know what else is semi permanent? Scarring. Be very careful before putting a soldering iron anywhere near someone else's wrist, because if you're not then they will probably end up with a different kind of permanent marking than intended.

Alternatives

Instead of welding the bracelet closed, if you're using something with links then you can just close the loop on a normal link instead of a catch using a pair of pliers. This would be just about as strong as welding it (because the other links in the chain won't be welded either), but avoid the need for extreme heat or expensive tools.

If you're still concerned about the strength of the link, you can apply a little bit of glue to it to make sure it says closed.