Button was running 36/10/1.13 over 110 hands. In the only notable hand that I played with him, he called a standard raise with KQo in the BB, check/called a Q hi flop, checked a flush draw on the turn, and then bet OOP when he rivered his flush.

What can he have that I beat?

PokerStars No-Limit Hold'em, $0.50 BB (6 handed) - Poker-Stars Converter Tool from FlopTurnRiver.com

CO ($60.45)
Button ($87.75)
SB ($36.20)
Hero (BB) ($49.25)
UTG ($30.85)
MP ($6.75)

Preflop: Hero is BB with 8, 7
3 folds, Button calls $0.50, SB calls $0.25, Hero checks

Flop: ($1.50) 3, 8, 7 (3 players)
SB checks, Hero bets $1.25, Button calls $1.25, 1 fold

Turn: ($4) 3 (2 players)
Hero bets $3.50, Button raises to $10, Hero ...?