Robert Bryant's comment motivates me to mention the "weighty" historical monograph Emergence of the Theory of Lie Groups (Springer, 2000) written by Thomas Hawkins. As usual with terminology such as "weight", the history reaches back into nineteenth century's invariant theory (Cayley, G. Kowalewski) but becomes most relevant to modern Lie theory in the work of Elie Cartan about a century ago. The early part of Chapter 8 in Hawkins' book is most pertinent but not easy reading.
Though Cartan's use of the term poids (weight, Gewicht) was not the earliest mathematical occurrence, it does seem to have been the first use in connection with what we now call weights of representations. There is also a long history involving the term "root" (and its offshoot "secondary root"), going back to antiquity, but here the work of Killing anticipates Cartan's more definitive treatment of semisimple Lie groups and what we now call their Lie algebras.
The history is not at all easy to untangle, but I think Hawkins was thorough in his study of the development of ideas along with terminology.
Terminology in this particular subject should not be taken too seriously, I think, and sometimes the names given to things are either misleading or inappropriate (including concepts named after people). Still, we are stuck with the language, which is almost impossible to change.