The phrase a bone of contention refers to something that causes a disagreement or argument, commonly seen as a reason for a quarrel.
quarrelsome: This refers to a person who is inclined to engage in arguments, but not the meaning of the phrase.
with a reason: While the phrase is related to a reason, it doesn't directly imply just having a reason; it’s about the subject causing the issue.
reason of quarrel: This is the correct meaning, as "a bone of contention" refers to the cause or issue that leads to a dispute or quarrel.
a disturbing man: This option is unrelated to the phrase's meaning, which focuses on an issue or point of disagreement, not a person.
\[
\boxed{\text{reason of quarrel}}
\]