Other people have answered your question, but I would like to make a suggestion to help you get motivations for your party!

In my campaign, each player will be part of the Grey Hands and one faction that is unique to them. The level 5 mission from the unique faction will be a hook to send them into Undermountain.

Examples: Bregan D’aerthe- Jarlaxle seeks to join the Lord’s Alliance, whether he gets the dragon horde or not. As a sign of good faith to his secret ally, Lady Silverhand, he will send his favorite new agent to investigate rumors of a dangerous war between two drow houses deep below in Undermountain.

Emerald Enclave- Jeryth senses a great evil power casting its tendrils of evil influence from deep below and sends her warrior to investigate and end the threat after granting him a vision of a grey and twisted version of Waterdeep. This is Alterdeep and the power below is the Ulitharid.

Etc...