What is Jerome saying with the tuft of hair?

A lock of hair is traditionally a token of remembrance, but throughout the movie, Jerome was giving his hair to Vincent purely for practical purposes. At the end, the hair was finally and purely given for its sentimental value. This symbolically fixed one of the broken things in the movie's dystopia.

Because real Jerome could never travel to Titan, by saying "I'm traveling too" I think it's obvious that by giving him his hair he, real Jerome, gets to travel to Titan as well. If you remember, he was curious "what is Titan like" and knowing he wold never make it there himself, by having his hair with Vincent, he vicariously gets to go to Titan, as well. Just my view.

It's been a while since I saw the movie, but apart from the sentimental interpretations in the other answers, I always considered the purpose of the hair to be more practical.

Throughout the movie Jerome allows Vincent to assume his identity by various means. This allows Vincent to fulfill his dream and Jerome to have some kind of fulfillment in his life as well. Jerome decides to end his life on this high note, but he does not want to end the dream of Vincent with it. So, he leaves him some more hair, so he can continue assuming Jerome's identity even when Jerome is no longer to support him directly anymore.

In a way Jerome is giving Vincent his blessing and the most precious farewell gift he has to offer: his identity. Note that Jerome has burned his remains, so Vincent will be able to return and live as Jerome.

Vincent has a knowing look after he opens the envelope with the lock of hair. I think Vincent knows that Eugene plans on suicide. First you have to find the links between the characters. Both Vincent and Eugene were not valued. Eugene isn't valued anymore because he is just "good genes" and achieved less than his potential. This is suggested by his name change from Jerome to Eugene. Eugene kills himself because he doesn't feel he has anything to offer. Vincent remembers Eugene saying "I'll be traveling too" and recognizes that Eugene has essentially linked himself with the better man. Eugene isn't necessary because he recognizes that Vincent with his dream and value in determination is better than all of his own genetics and value in potential. It was his way of saying I'm with you in spirit and ideology.

Another option is that the lock of hair is from Vincent original, and was given to Jerome when they replaced identities. Vincent was trying to look (and be) Jerome but Jerome also needed to apper to be Vincent if the accation apper itself.

Jerome returning the lock of hair to Vincent is like saying - it's really you that sould be up there - you are the better man. And it's a farewell as well - for he won't be needing it any more.

Basically I think Eugene thinks he hasn’t had anything else to offer. When he gave Vincent his hair it’s like he was traveling too. But also pay attention to why Eugene is actually in the wheel chair. It was never an accident. He says he tried to jump in front of the car. But Vincent thinks that he’s just drunk. However this could be his second attempt at suicide. Before even when he had his genes and his legs he wasn’t happy. Showing that each character intertwined. Eugene wanted a normal life where he wasn’t expected to be perfect because of how he was born. But Vincent wanted to be recognized because he was perfect. He wanted a goal that couldn’t be reached with his genes.