I think you played the hand just fine. I agree with cactus that checking the flop was fine. If you do decide to lead out and face a good sized raise from someone in position, you are going to be in a bad spot and have to make some tough decisions throughout the hand. The min raise on the turn does seem strong but he is giving you over 3.5 to 1 pot odds
so you are calling every time. Once you make your second pair on the river you are strong enough to call him down. I agree with cactus that the villain could be holding 99 or a less likely straight. However, I can also see your opponent playing this way with lots of hands that you beat such as QT, KQ, AQ, TT, JJ, KK, AA and others. There are just too many hands that you beat.....You gotta call. Good Luck.