It is not known why Dirichlet denoted his functions with an $L$. Perhaps he chose $L$ for Legendre (I am not serious). The reason may be alphabetical. Just before $L$-functions are introduced in his 1837 paper on primes in arithmetic progression (Math. Werke vol. 1, 313--342), there are certain functions $G$ and $H$, and the letters $I, J$, and $K$ may not have seemed appropriate labels for a function.

While $L(s,\chi)$ and $L(\chi,s)$ are common notations for the $L$-function of a character $\chi$, neither decorated notation is due to Dirichlet; he simply wrote different $L$-functions as $L_0, L_1, L_2,\dots$.

Update (Jan. 12, 2016): I learned a few days ago from Ellen Eischen that the Kubota Tractor Corporation has a model called the "(compact) Standard L-Series," and today I saw a Kubota L-series go past my department building. Here is a photo I took.

If you're looking for a modern reinterpretation of what the L stands for in L-series, the webpage https://www.kubota.com/product/tlbseries.aspx gives the answer, and it's not Langlands: L means Loader or Landscaper.

someletter. He used L. Whatever he had used---would you have asked what the reason was? Why do number theorists use T for Hecke algebras? It's just what someone chose and it stuck. Itmightbe no more than that... – Kevin Buzzard Apr 14 '10 at 19:37