Why is he so important?
For multiple reasons. Jeff's death feels heavier in the adaptation -- for Hannah, Clay, and viewers -- because everyone's more invested in him. He's no longer a random, nameless student. He's Clay's friend, the catalyst who sets TV Clay and Hannah in motion. In fact, multiple Clay-Hannah moments -- their first dance and kiss, in particular -- don't exist without his encouragement. ("The only time you and I actually ever get to hang out is because of Jeff?" Hannah points out.)
In a show that hits on numerous hot-button teen topics -- being kind to others, talking about your feelings, and asking for consent among them -- Jeff's subplot also adds a look at the real ramifications of drunk driving. (Even if Jeff wasn't driving under the influence, was Hannah's driver?) The show tries not to be too on-the-nose with its preaching, but echoes of "designated drivers" and "how much have you had to drink?" send a clear message.