So an elven monk wants to wield a long sword one handed. Does that count as a monk weapon?
If you and your DM/table agree that it does? Absolutely. Sure, the PHB says no, but the longer you play the game the more you should develop a healthy disrespect for the PHB. As they say,
the map is not the territory.
What is the intent of the Monk weapon restrictions?
Primarily, it's to limit players to selecting weapons that reinforce the flavor of the class. That flavor is an eastern style martial artist, which developed primarily using weapons available to peasants. However, styles exist that employ twin swords (twin butterfly swords), sabers (dao), and what would reasonably be considered a longsword (jian). Mechanically, however, it restricts you to "weapons that are do not have the two-handed or heavy" primarily to prevent the player from benefiting from the higher damage dice with the extra attacks that the Monk class benefits from.
What are the outcomes of allowing a longsword to be a Monk weapon?
1. Well, you kind of get about +1 damage per attack, since most Monk weapons deal 1d6 damage. Quarterstaff and spear do 1d8 with versatile, but I know some tables don't allow that. If your table is the type of table that's going to get upset about the extra damage, you'll want to take that into account. It sounds like you've already done that partially by denying versatile from longsword.
2. If you use random treasure, your Monk player might have a slightly better equipment draw. However, if you're going to give out magical weapons as treasure, most DMs tend to give players something they can at least use.
That's... really it. If #1 is *really* a problem, then let them use longswords with shortsword or spear statistics. If #2 is *really* a problem, then don't use random treasure, or tell the player that they lose proficiency with everything except dagger and longsword due to the unusual focus of the school. That's not a significant penalty as far as weapon selection, but it does give them a much narrower equipment draw.
Personally, I'd let the player have it, but help the player develop a backstory: "Your master, and elf known to wander the wilderness, eventually diverged from the standard teachings. In the vast tundra of the frozen north, he came across a crippled silver dragon fighting seven frost giants. Alone with a mangled wing and a misshapen jaw, the dragon killed all seven giants while protecting itself using nothing but it's tail. He aided the dragon and nursed it back to health as best he could, but the dragon's injuries were not new, and not within your master's ability to cure. After years of study and meditation, he used the dragons way of fighting to develop The Way of the Silver Dragon Tail. Some schools will shun such a technique, and may distrust or fear you as a result. Additionally, your master taught you to be proud of his teachings and the style. You are generally unwilling to use any other weapon or form."
