The simple answer: The neutrality of Switzerland.

Neutrality is the cornerstone of the Swiss foreign policy. It has not participated in a foreign war since 1815 when the Treaty of Paris restored the Swiss Confederation. While, the last time that the country participated in a war of any kind was its own civil war, Sonderbund War of 1847.

Switzerland remained neutral during World War I and this neutrality was recognised by both the Entente and Central Powers.

The choice of where to locate the League of Nations, the predecessor of the UN, came down to two cities: Brussels of Geneva. As the League of Nations was the brainchild of President W. Wilson, he arrived at the decision of Geneva.

According its successor organisation, the UN, President Wilson favoured it:

primarily because of Switzerland’s neutrality. He felt that if Germany ever did join the League of Nations, it would be a far more acceptable place because the painful memories associated with Belgium could be avoided.

In 1943, the 3 powers (U.S., the Soviet Union and the UK) decided that a new organisation ought to replace the League of Nations. It was presented by Roosevelt to Stalin and approved by Churchill.

During the World War II, many of the League's affiliated bodies continued to function including the International Labour Organisation and the Health Organisation and these eventually became affiliates of the United Nations. Although, the Health Organisation became the World Health Organization.

At the end of World War II, all League of Nations' assets in the form of buildings, properties (Le Palais des Nations, the HQ of the League included) and archives were transferred to the United Nations. Sean Lester, the last Secretary-General of the League of Nations signed the final act of transfer on 18 April 1946.

The United Nations administration still is based in Le Palais des Nations along with the United Nations Conference on Trade and Development (UNCTAD), the United Nations Office for the Coordination of Humanitarian Affairs (OCHA) and the United Nations Economic Commission for Europe (ECE).

Sources:

United Nations Office at Geneva (2005) Organization and establishment. Available at https://www.unog.ch/80256EDD006AC19C/(httpPages)/DC4AA21261EF9EE4C1256F31002C7B73?OpenDocument (Accessed: 30 November 2018)

Mc Carthy, M. 2002. A brief history of the World Health Organization. The Lancet. 360(9340), pp. 1111-1112.

United Nations Office at Geneva (2005) End of the League of Nations. Available at https://www.unog.ch/80256EDD006AC19C/(httpPages)/BA9387B56BFAAFB4C1256F3100418C75?OpenDocument (Accessed: 30 November 2018)

Mc Namara, P (2009). Sean Lester, Poland and the Nazi Takeover of Danzig. Dublin, Ireland: Irish Academic Press.

*edited to include more sources as I thought that the question asked referred to the League of Nations only