Why are they called L-functions?

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.

Many have suggested that it comes from "Lejeune", as in "Johann Peter Gustav Lejeune Dirichlet". I have never seen this properly sourced and have often wondered if the claim is legitimate.

Whatever the historical reasons are, I think it is a good thing to use the terminology 'L-function' because of Langlands's amazing contribution to the theory of automorphic forms. Moreover Langlands functorialities are stated in terms of the 'L-group'.