My personal headcanon is that this is when the WoL sits down and researches the spells or trains manuveurs that he "learns"/masters when he levels up.
Then there's simple stuff like cleaning your private room at the FC house, keeping the flowers watered, grooming your chocobo,... Regular people stuff.
When not actively MSQ-questing, the WoL also likes to just sit back in the Waking Sands/Rising Stones/Fortemps Manor/Moghome/Dragonhead/Whitebrim Front or whatever other place he's currently visiting. Talk to the locals, visit the market, train with the soldiers, have meals with the current head honcho, discuss arcanima theory/conjuration with the recident healers...
There's a reason why, that as you progress through the MSQ, people from stable hands to royal guards go from calling the player 'you', to speaking to him/her like like they know you at least as a casual acquaintance.
Because despite appearances, the WoL is pretty darn social... Everyone from 2.0 Alphinaud to the Scions to faction leaders to Gaius van Baelsar himself seems to have no problem getting along with him/her.