BB is 45/24/44% (AFq) ~ 60 hands. 3b is at 5% so far.

Info from hands I've seen him SD thus far:

Preflop
Likes to Open mini-raise: ATo (SB), 87s (SB), QJs (CO), K9o (BTN)
He mini-4bet the QJs vs. a BTN mini 3-bet and called a 5-bet
Open limped Q9o (CO)

Postflop
He played top 2-pair HU fast on the flop vs. a 40bb SS
He c/c x 2 with a FD OOP and did not attempt to bluff the river w/ he missed (had no SD value)
Peeled the Q9o hand above on the turn (limped pot with flop getting checked through) on KJ43 2T board
Did not c-bet K9o on QJJ 2T flop HU, min-bet turn A and river 9 when checked to.

Poker Stars $0.10/$0.25 No Limit Hold'em - 5 players
The Official 2+2 Hand Converter Powered By DeucesCracked.com

Hero (UTG): $24.65
CO: $25.00
BTN: $27.00
SB: $25.35
BB: $18.90

Pre Flop: ($0.35) Hero is UTG with J J
Hero raises to $0.75, 3 folds, BB raises to $1.25, Hero calls $0.50

Flop: ($2.60) 8 2 4 (2 players)
BB bets $1, Hero calls $1

Should I have raised here? My thinking was he would likely fold to a raise if all he has is whiffed OC's whereas he could continue to barrel if behind.

Turn: ($4.60) Q (2 players)
BB bets $2.30, Hero calls $2.30

This card probably hits part of his 3-bet range. I decide to call again to keep the pot small and reeavluate on the river.

River: ($9.20) 3 (2 players)
BB bets $3.75, Hero?

OK, I got thus far and he barrels again for just under half the pot. Do I call?? This feels like a fold. I think he would have bet smaller with weaker hands.

call. hard to tell but it looks kinda like a series of blocking bets or discount barrels and could very well be a smaller pocket pair. plus you've underrepped your hand the whole way.
I like a reraise preflop (to set up a postflop jam) provided he calls wide range preflop.