There have been a few threads regarding making PCs out of other races. While Warforged and Kender characters are slated for the DMG and lycanthropy and vampirism are mentioned in the MM I believe, there are not the notes that the 3.5 MM had to make a PC out of almost any race.
My suggestion would be for the DM and player to work together to come up with a story based reason for this PC of another race. Then they need to figure out what the racial traits and abilities would be.
All non-human PCs should have +2 increase to 1 ability and +1 to another. The MM will give you stuff like speed, darkvision and suggested alignments. It also will give ideas of which abilities to increase (choose the top two abilities in the stat block) as well as what features they have.
We have a half-gnoll paladin in our current campaign. He is the child of the player's human druid from our last campaign and a good aligned gnoll. We gave him +2 str/ +1 dex, darkvision, a bonus action to attack another enemy when he knocks an enemy unconscious and proficiency in intimidation from the half-orc page. Given all of that, it is still less than what most PC races in the PHB get. We decided to stop there so that gnolls did not become too awesome and get in the way of the story. The player still does a great job of playing up his gnoll characteristics, but no one is jealous of him because he gets better stats.
I would also recommend sticking with medium and small creatures until we get more info on tiny and large creatures. You can search for the minotaur thread to look at what we discussed about that kind of PC.