I've tried to search on Google for the etymology of this word but I couldn't find it (or at least I couldn't understand that it was an etymology since it was written in German).
Bedürfnis should mean something that I need – a desire (etwas was ich brauche).
The strange stuff is that the be- prefix in German is often used to handle the accusative (correct me if I'm wrong) but the verb dürfen means something like "I can".
Bedürfnis is a derivative of bedürfen with the suffix -nis. According to DWDS (bedürfen) and DWDS (dürfen) the root of bedürfen and dürfen is
Old High German: thurfan / bithurfan = in need of something or in general
The meaning of the prefix be- is unclear in that setting.
One of these two, thurfan, developed into another meaning:
Middle High German: durfen = having reason
New High German: dürfen = having the right to
while the other, bithurfan, kept its meaning.
"einer Sache bedürfen" means to need sth. As the verb is used with a genitive object it is on the wane in written language. It is no word of colloquial language. There are two nouns, Bedarf and Bedürfnis. The etymology of these words does not help much. The older wordforms don't shed more light on these words than that they express need or what is necessary. Bedarf is a current word today:
der Bedarf an Rohstoffen / Erdöl/ Energie. Bedürfnis is not so common. It is used in polite formulas such as
Es ist mir ein Bedürfnis, Ihnen meine Bewunderung auszusprechen.
Sometimes you find an older name for public lavatory/WC/toilet: Bedürfnisanstalt.
