The correct answer to this open question is the following.
The role did the Oregon issue played in the 1844 presidential election was the following.
The Democratic party insisted that the United States had a genuine claim to demand Oregon as part of the US. At that time, the British controlled Oregon. This issue met the other territorial expansion demand: the annexation of Texas to the Union. That is why the Democratic candidate James Polk included in his platform the expansionist idea that later became the Manifested Destiny, the right of the US to expand and grow. Polk won the presidency by defeating Henry Clay, who opposed the annexation of Texas.