My first suggestion would be to tell the player, "Sorry, no. Not this time." While this sounds like being a bastard DM, let me explain.
1) You've already said you are trying to get your head wrapped around the races that exist. Why does he/she need to choose/play a race that is NOT presented in the book already?
2) If this is to be your first 5e game, and it sounds like it may be, you have plenty of stuff to cover/get used to without making things up out of the gate...which will likely have to be revised anyway when WotC puts out some new manual (maybe even just the MM or there might be "bad guy" races in the DMG) with their "official" duerger in it. Again, why is this particular race choice necessary at this time?
3) Presuming you will be running a "typical" D&D 5e world, it's established that these guys are Underdark guys. Are you running an Underdark campaign? Are you running an "evil" party/campaign? Then the choice makes, at least some, sense. As a "well it's the one thing not listed so it's automatic different/special snowflake/I wanna run a Dwarven Drizzt" it doesn't make sense for this particular single duerger (who will be reviled, feared, and/or likely attacked on sight by any mountain or hill dwarf it encounters) to be wandering around the upper world.
Now, considering you seem at least willing to give this character a shot, I suppose you should follow the basic set up of the other races.
I would use Mountain Dwarf as your base for Ability mod's. )+2 Con. +2 Str.), Age, [Societal] Alignment: Lawful Evil, Size, and Speed. Use the standard Dwarven Resilience, Stonecunning, Dwarven Armor Training as is.
Tool Proficiencies with smith's tools, mason's tools, poisoner's kit.
Combat Training with throwing hammer, short sword, dagger and crossbow.
Languages: Common (for ease of play/interaction with the other characters), Dwarvish, Gnomish, and either "Underdark Common" [if you are going to have such a thing] or Elvish [specifically Drow speak if you use a different language for them].
Darkvision enhanced out to 120' (that's what the Drow have).
Sunlight Sensitivity: this is a biggie must if the player wants to use this race.
Duerger Magic: [using what have been traditional duegar abilities and what was specifically said of them in the PHB...which gives them greater innate magic capabilities than other PC races, btw] You know the "Ray of Frost" cantrip, refluffed as a psionic attack/psychic damage. (Effects/duration/etc. are the same. The visual of a stunning mind blast is up to you/the player.) At 3rd level you can cast Invisibility on yourself once per day. At 5th level, you can cast Enlarge, on yourself, once per day. Intelligence is your spellcasting ability for these spells.
That should suffice to get the ball rolling.