Why is Shimano's 105 groupset called 105?
This is a non-answer because I've been looking at Japanese bicycling sites and blogs and no one really knows. The closest answer is found here:

105の名の由来とは？
最後に、各グレード名の由来についての豆知識を紹介します。1971年に誕生したデュラエースの名は、素材のジュラルミン（Duralumin）と耐久性を意味するデュラビリティ（Durability）に、世界一のコンポという願いを込めた「エース」を組み合わせた造語です。1976年に誕生のアルテグラは、Ultimate（究極）+Integrate（統合）を意味するコンポーネントです。1982年に誕生した105については、当時のダブルレバーSL-A105の型番を継承し、現在では105というグレード名になっています。

which google translate translate (with some clean-up) as:

What is the origin of the 105 name?
Finally, I will introduce my knowledge about the origin of each grade name [in the Shimano series]. The name of Dura Ace which was born in 1971 is a combination of Duralumin of the material and Durability meaning durability combined with &quot;Ace&quot; which wish of the world's best component. The birth of Ultegra in 1976 is a component that means Ultimate + Integrate. Concerning 105 which was born in 1982, it inherited the model number of the double lever SL-A 105 at that time and now it has become the grade name of 105.

So it's a non-answer answer. The author of this article, Kenji Hashimoto, is a sports journalist in Japan and so it's about as close as one might get unless you ask Shimano.