I have an elf in my group who only needs 4 hours of rest instead of 8 like all the others. The problem is he keeps asking what he can do in that time and I have no idea what to tell him. I've read in the PHB about crafting and such but it doesn't really flesh out how the system actually works. Can he for example make potions? Craft weapons? So far he has done miscellaneous things like perform for money and hunt for herbs (which I have no idea what they do).
It's important to note that the Elf's 4 hours of sleep grants 8 only absolves it of the need to sleep, not of the need to rest 8 hours (or maybe it doesn't, that debate rages here).
So depending on which interpretation of the rule you decide to use for your game (neither one is game breaking, so it's really a pick-em for the DM), you can then use that to determine the kinds of actions that are allowable during the remaining 4 hours.
If you use the looser version of the rule things like scouting, foraging, and even short jaunts into nearby settlements are completely fair game. It's worth spending some time out of game working with the player to figure out what kinds of things he wants to do during this time. However, I'd advise against taking up game time dealing with this on a regular basis as it's something that concerns one single player and most times will be irrelevant to your other players.
If you take the designer's interpretation of this rule (and really in a mixed party, I think this is the one I'd prefer), things like taking extra watches, transcribing scrolls into spell books, brewing potions, cooking the party's meals, are all things that would be good uses of that times and great services to the party.
The important thing here is to provide the racial trait with weight, so it feels special, but doesn't make the character seem overpowered. I feel like that is what the designer's are trying to prevent with the sleep vs rest rule, they're trying to prevent the elves from feeling overpowered (I doubt very much that this would actually lead to any real balance concerns, but feel is important).
In our group, the elves take extra turns on the watch. This benefits them as they are about twice as likely to be on watch and get an extra round of combat if there is an encounter (instead of needing to wake up and grab weapons). It is also very simple.
The player is looking for other ways to leverage his character's racial advantage. That is understandable, but it may distract from the main plot, because other characters cannot take part.
I suggest leave it to the player, there is no need to feed them extra "exciting" game-playing opportunities. In fact that will almost definitely cause distractions. So just say "yes" to all the minor roleplaying bits and pieces the player wants to do with the PC's spare time, but avoid turning it into long solo sessions. Hopefully when you put the group into a more hostile environment, the group will figure out the watch advantage and the player will happily be the group's hero by taking an extra watch (in the hope of getting the jump on any action).
When there aren't many threats around, treating the time as "downtime" for scroll-transcribing or potion making may be fine too (although do note that by RAW many such activities require 8 hours per day dedicated, so either the elf has to find another 4 hours somehow, or you might use DM discretion to over-rule RAW where 4 hours-per-day but half the speed seems entirely reasonable and in the spirit of the game). Anything which can be described and resolved in a couple of minutes is what you should aim for.