So in this thread I will list important dates and events regarding the PokéGear phone:
On
Sunday morning you may re-battle
Lass Kriste by calling her.
On
Sunday evening you may receive a
Leaf Stone by calling
Picnicker Gina.
On
Sunday evening you may receive a
Water Stone by calling
Fisherman Chris.
On
Sunday night you may re-battle
Picnicker Gina by calling her.
On
Monday morning you may re-battle
School Kid Jack by calling him.
On
Monday morning you may re-battle
Pokémaniac Brent by calling him.
On
Monday evening you may re-battle
Youngster Joey by calling him.
On
Monday evening you may re-battle
Firebreather Walt by calling him.
On
Tuesday morning you may re-battle
Bug Catcher Arnie by calling him.
On
Tuesday evening you may re-battle
Picnicker Tiffany by calling her.
On
Tuesday evening you may re-battle
Gentleman Alfred by calling him.
On
Tuesday night you may re-battle
Bug Catcher Wade by calling him.
On
Wednesday morning you may re-battle
Fisherman Ralph by calling him.
On
Wednesday afternoon you may receive a
Fire Stone by calling
Schoolkid Alan.
On
Wednesday night you may re-battle
Bird Keeper Vance by calling him.
On
Wednesday night you may re-battle
Sailor Huey by calling him.
On
Thursday morning you may re-battle
Fisherman Wilton by calling him.
On
Thursday evening you may re-battle
Picnicker Liz by calling her.
On
Thursday night you may receive a
Thunderstone by calling
Lass Dana and defeating her in battle.
On
Friday morning you may re-battle
School Kid Chad by calling him.
On
Friday evening you may re-battle
Hiker Parry by calling him.
On
Friday night you may re-battle
Hiker Anthony by calling him.
On
Saturday morning you may re-battle
Camper Todd by calling him.
On
Saturday morning you may re-battle
Youngster Ian by calling him.
On
Saturday night you may re-battle
Picnicker Erin by calling her.
JUGGLER IRWIN
His battle statistics vary greatly, therefore he needs his own section. His availability resets each week. All timings based on when you interact with him on Route 35 (after having called him).
In Winter, he only gives you a Leaf Stone. Will battle you first.
In Spring, speaking with him during the day will allow him to give you a Fire Stone.
In the Fall, speaking with him during the day will allow him to give you a Sun Stone. talking to him during the night will allow him to give you a Moon Stone.
REQUIRES FURTHER RESEARCH...
This thread will eventually be extended, now I need to get back to school work. 
