Being primarily a machinist rather than a welder, I would make a new pin from scratch. While you may succeed in a welded pin working, the new pin will have better mechanical properties that make is less likely to fail again. Keep in mind that a weld is essentially a casting. Castings have grain size and alignment that is not ideal for strength or toughness. On a firing pin, there is no option to make it thicker in crossection for added strength to compensate. Typically, firing pins are made from bar stock. Bar stock has mechanical properties similar to a forging where the grain size and alignment is more idealized.
Another option that I've seen on this board is the drilling out of the tip and pressing in of a new tip.
No matter if you weld it, drill and replace the tip or make a new complete pin, it needs to be of the correct hardness, which often means heat treating.
