Interesting question. Apparently to understand why 青 is used in 青年 we have to go look at 青春{せいしゅん}. The origin of this word is closely related to the Chinese concepts of Yin/Yang and Wuxing (the five elements).
Basically you have the five elements: 「木・火・土・金・水」each of them connected with a cardinal point, color, time, and god.
In particular the wood (木) is related to east (東), to 青 ("blue" as you said but notice that another meaning for it is young/immature. More on the color later), spring (春), and 青竜 (the blue dragon, an auspicious mystical creature said to rule over the eastern heavens). Among these 青 is the color of spring and hence 青春 became a synonym of spring.
Now from here the connection is straightforward since 青春 is itself a pseudonym for the early years in one's life.
The precise reference of this part is here:
春の異称「青春」が、年の若い時代をさすようになったのは、夢や希望に満ち溢れていることから、人生の春にたとえられたものである。
Therefore, connecting the dots, most likely 青 in 青年 comes from 青春 and the meaning related above.
You can find the full quote above from the entry for 青春 in an etymology dictionary here. For convenience I'll copy it here too:
青春は、陰陽五行説に由来する。
陰陽五行説には、「木・火・土・金・水」の五行があり、各々に対応する「方位」「色」「時」「神」がある。
五行の「木」は、「東」「青」「春」「青竜」に対応し、春の色は「青」であることから、「春」の異称が「青春」となった。
「朱夏」「白秋」「玄冬」も、陰陽五行説に対応する色に基づいた季節の異称である。
春の異称「青春」が、年の若い時代をさすようになったのは、夢や希望に満ち溢れていることから、人生の春にたとえられたものである。
季節の異称の中で「青春」のみが人生の時期を表しているのは、「青二才」や「青臭い」のように、「青」に「未熟」の意味が含まれていることも影響していると思われる。
Here is the definition of 青年:
青春期の男女。10代後半から20代の、特に男子をいうことが多い。若さを強調する場合には30代にもいう。わかもの。わこうど。「青年実業家」
A note on the color 青
As pointed out in a comment, let me give some more info about the actual color of that 青 represents.
Let's start off by saying that the translation "blue" is an over simplification. Of course according to wikipedia, that's the most natural translation one would give (as it is listed as one of the three primary colors). Also, its RGB code is #0067C0 which corresponds to blue indeed.
However, in general it is used in every day life to refer to a wider spectrum. One interesting example is for sure the color of traffic lights. What is called 青 in Japanese is what any westerner (I guess) would call green (reference pic below). Now, actually I believe that some Japanese people as well might argue that the traffic light color is actually "green", but the point of this example is just to stress the ambiguity of 青 and show that (especially in daily life) it might represent a wide spectrum of colors rather than a single specific one.
As I also mentioned above, it is considered the color of spring, hence would be not surprising if green comes into play as well. In this case, it is interesting to notice the association that the color green has in English with youth and inexperience. See here for example.
