Was the character a Barb first or a Warlock first? By your title, it looks like barb first. At what level did the MC start?
1) Will this player also want to cast spells or cantrips while raging?
2) How, if at all, would this affect the advantage the player gains on Strength checks and saves while raging?
I wouldn't allow it. It's a MAD build; those exist. It will work fine putting points into both Str and Wis.
On a related note, it seems like a "Totem Spirit" patron could be a great subclass.
Hex Warrior says "you can"... just use your Str/dex stat.
Was the character a Barb first or a Warlock first? By your title, it looks like barb first. At what level did the MC start?
If a Barb first, then str is the main stat from the beginning and to make a change part way through would take some doing, RP wise... maybe a generic feat to show training to use emotion(chr) as power rather than str.

(Dungeons & Dragons)
Rulebook featuring "high magic" options, including a host of new spells.