As suggested by Gerald, the notion was first introduced for groups. Given a directed system of groups, their direct limit was defined as a quotient of their direct product (which was referred to as their "weak product"). The general notion is a clear generalization, although the original reference only deals with groups. As mentioned by Cameron Zwarich in the other [answer][1], the definition can be traced back at least to Lev Pontryagin.

For an early exposition in English, see 
> [MR0007093 (4,84f)][2]. Lefschetz, Solomon. **Algebraic Topology**. American Mathematical Society Colloquium Publications, v. 27. American Mathematical Society, New York, 1942. vi+389 pp.

Specifically, direct limits are defined in Chapter 2, $\S$14 (p. 57).


  [1]: https://mathoverflow.net/a/357967/6085
  [2]: https://mathscinet.ams.org/mathscinet-getitem?mr=7093