The simplest weapons, such as the spear (including the javelin), the club, the thrown stone, are all simple enough to have easily been invented multiple times. They are also exceedingly ancient inventions. Spears have been used for at least a few hundred thousand years, and probably much longer. Observation of chimpanzees making and hunting with spears:

• Pruetz J. D., Bertolani P., Ontl K. Boyer, Lindshield S., Shelley M. and Wessling E. G., (2015), "New evidence on the tool-assisted hunting exhibited by chimpanzees (Pan troglodytes verus) in a savannah habitat at Fongoli, Sénégal", R. Soc. open sci. 2, 140507. http://doi.org/10.1098/rsos.140507

suggests that they might have been used by the common ancestors of chimpanzees and humans. Noting that no other chimpanzee groups have been observed hunting with spears, it's likely that this observed use is itself an independent re-invention of the spear. Clubs are used much more widely by chimpanzees, and also the throwing of objects.

For more complex weapons such as the bow, the sling, and the spearthrower, we don't know. They are all very old, very prehistoric inventions. Since they are made (mostly) of wood and fibre, they survive quite poorly, and their early history is mysterious. If the bow was invented at about the time that the oldest surviving fragments date to (about 10,000 years ago), then the bow was certainly invented multiple times, with at least two independent inventions (the Old World and the Americas), and probably more. With the earliest dates for which we have evidence (about 65,000 years ago), in the form of projectile points which, if made in the last few thousand years, would have been identified as arrowheads, the bow might have been invented once, and spread. Even with this early invention, it might have been invented multiple times. One version of the bow, the wood-horn-sinew reflex-recurve composite bow has two independent inventions, one in Central Asia probably about 4,000 years ago, and the other in America.

The crossbow might have multiple inventions (possibly in eastern Asia, Western Asia or Europe, and Africa), or possibly just a single invention - the early development and diffusion of the crossbow (as a hunting weapon rather than a military weapon) is almost unknown. The development of powerful military crossbows occurred multiple times, at least twice, in East Asia and Europe.

Metal daggers or short swords have been invented at least twice, possibly many times. There was at least one Old World invention and at least one American invention (and probably at least two American inventions, one in the Pacific NW, and once in the Andes, and possibly also Mesoamerica). Similarly, metal axes have at least two separate inventions, at least once in the Old World and at least once in the Americas.

In summary, we don't know whether bows (in general), axes (in general, including stone axes), and spears were invented multiple times. Spears are simple enough to have been easily invented multiple times, but we don't know if they were. Later developments, such as military crossbows, composite bows, metal daggers/swords and metal axes, were invented independently multiple times.