Using downtime activity to research spells

I run a D&D 5E campaign. I have a player who is playing a wizard. This player would like to use his downtime to learn new spells. Specifically, he doesn’t want to create any new spells of his own, he wants to use his downtime to write additional spells (from the PHB) into his spell-book. The PHB and DMG do not provide rules for how to handle this.

This player is both a self-professed rules-lawyer and self-professed optimizer; he is also the only one of each in the entire group. Because of this, it is very important to have a set of rules that are clear and unambiguous, but don’t set his wizard up to being over-powered compared to his party-mates.

How do I allow my player’s character to spend his downtime learning new spells without allowing the character to become overpowered?

A good answer will provide a set of rules or guidelines that allow my wizard player to spend his downtime adding new spells to his spell book. The rules must:

  • be clear and unambiguous
  • balance time and cost in a way that keeps this downtime activity in line with other downtime activities
    • This means that the rules should not be so constraining that spending downtime in this way feels useless, but doesn’t allow the wizard to become over-powered by having access to too many spells.
  • have been used and put in-practice (in a 5E game), and the accompanying experience using the rules should be included
    • I’m not looking for, “I think you could do it this way”; I need, “We did it this way, and here’s how it worked for us”.

The very best answer will do the above while making it clear how this will not unbalance the party (Cleric, Ranger, Rogue, Monk, Warlock).