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The wizard in my campaign has the Eye of Vecna and is a yuan-ti, and this means he has suggestion and dominate monster at least once a day. Could he simply suggest to a demon that it reveal its true name? Or force it to?
If so, wouldn't that effectively give him permanent power over the demon, and make a spell like planar binding basically useless?
Yes, you can learn a demon's name this way.
Dominate monster could be used to compel a demon to reveal its true name, but the demon gets a saving throw (and most demons have Magic Resistance, so they will have advantage on the saving throw). Since revealing its true name is "obviously harmful" to the demon, suggestion would not be able to compel it in that way.
The most common use for a demon's true name is as part of the 4th-level summon greater demon spell (in Xanathar's Guide to Everything):
At the end of each of the demon’s turns, it makes a Charisma saving throw. The demon has disadvantage on this saving throw if you say its true name. On a failed save, the demon continues to obey you.
Gate can also be used to summon a specific demon whose true name is known — perhaps one too powerful to be summoned and controlled via summon greater demon:
When you cast this spell, you can speak the name of a specific creature (a pseudonym, title, or nickname doesn’t work). If that creature is on a plane other than the one you are on, the portal opens in the named creature’s immediate vicinity and draws the creature through it to the nearest unoccupied space on your side of the portal. You gain no special power over the creature, and it is free to act as the DM deems appropriate. It might leave, attack you, or help you.
Note that in the case of gate, knowing the demon's true name doesn't help you control it; likewise other spells like planar binding or, indeed, dominate monster are not more effective if the demon's true name is known.
This answer has been edited to include new information from Xanathar's Guide to Everything, which includes spells that specifically benefit from knowing a demon's true name.
Suggestion just requires a suggestion that sounds reasonable within a sentence or two. You could suggest the following as a very reasonable suggestion:
"I'll make you a deal demon. You tell me your true name so I can protect myself from you on the prime material plane, and I'll release you to wreak havoc on anything you please, excluding me."
On MM pg. 53, there's an excerpt about demons directly related to this, emphasis mine:
Demon True Names
Though demons all have common names, every demon lord and every demon of type 1 through 6 has a true name that it keeps secret. A demon can be forced to disclose its true name if charmed, and ancient scrolls and tomes are said to exist that list the true names of the most powerful demons. A mortal who learns a demon's true name can use powerful summoning magic to call the demon from the Abyss and exercise some measure of control over it. However, most demons brought to the Material Plane in this manner do everything in their power to wreak havoc or sow discord and strife.
However, this also highlights that you would only have some measure of control, not permanent power. The follow up sentence in the description provided highlights that the demon will actively fight you on this, making Planar Binding a requirement to force something into servitude.
On a personal side note: This is why I use Devils instead of Demons. Devils don't like service, but they expect to be rewarded and will willingly follow commands when they are issued. Source is the entry on Devils on MM pg. 66, under the section "Obedience and Ambition".
