One of your old favourite songs from way back when
For the fun of it, straight from the plugin page.
This is not just a plugin, it symbolizes the hope and enthusiasm of an entire generation summed up in two words sung most famously by Louis Armstrong: Hello, Dolly. When activated you will randomly see a lyric from Hello, Dolly in the upper right of your admin screen on every page.
A few interesting (and not so interesting) facts about Hello Dolly
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Has been downloaded more than 480 000 times, so there are people that actually uses the plugin
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Last update was in February 2012, will we ever see an update and will it survive in near future releases of WordPress?
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The current rating of the plugin is 3.3/5, which is above average.
The truth of it, “Hello Dolly” was included in version 2.7.1 and has become a feature ever since. Whether is was a practical joke from one of the developers of WordPress (the only way to really know is to contact the original guy who decided to include this plugin), or serves the purpose of demonstrating/showcasing the structure of a simple plugin, no one will ever really know, except that specific wordpress developer.
The sweet and short, delete it if it irritates you. No one is forcing you to use it. It really serves no purpose anyway