Would is used to convey the conditional mood. The conditional mood conveys some sort of dependency on a condition. A more obvious example:
If I had any sense, I would come in out of the rain.
In this case, it is fine to say The best choice is "Having finished eating ..." as well. Using the conditional conveys a sense of If you were to say this again, the best choice would be "Having finished eating ..."
Conditionals are often used in a sense like this, to "soften" the statement a bit. This is a good short overview of various forms of the conditional.
Now, would doesn't always denote conditional mood. It is also the past tense of will:
They asked me to work late tomorrow, and I won't (will not) do it.
They asked me to work late yesterday, and I wouldn't (would not) do it.
Would is also used in a past sense to convey being in the habit of doing something (similar to used to):
When I lived in Poughkeepsie, I would/used to take a walk every morning at eight.
There are other meanings, and you can see more of them here.