I have encountered what seems to be a bug where I receive no dialogue option to begin the "assist the people of the rift" objective. I completed the task the jarl set for me, and now she has nothing more to say to me. Can anyone help? Thanks in advance.

You cannot become the thane of Riften once Maven Black-Briar is Jarl. This means if you bought the house after she was made Jarl, no housecarl will be awarded and the room in the basemant will remain empty. ( Still found after Patch 1.5)

You will be able to buy Honeyside, but not named Thane. The quest "Return to Jarl" stays unfinished.
Workaround: Visit the Jarl to complete the quest before entering Honeyside.

Try helping/doing more quests for people in Riften and see if that helps as well.