tptk against a raise, check.

PokerStars No-Limit Hold'em, $0.02 BB (9 handed) converter

MP1 ($1.55)
MP2 ($1.44)
MP3 ($5.78)
CO ($6.34)
Button ($2.80)
SB ($0.69)
BB ($2.32)
UTG ($2.24)
Hero ($6.91)

Preflop: Hero is UTG+1 with A:heart:, J:heart:.
1 fold, Hero raises to $0.1, 2 folds, MP3 calls $0.10, 3 folds, BB calls $0.08.

Flop: ($0.31) J:club:, 7:diamond:, 2:heart: (3 players)
BB checks, Hero bets $0.2, MP3 folds, BB calls $0.20.

Turn: ($0.71) 4:spade: (2 players)
BB checks, Hero bets $0.4, BB raises to $0.8, Hero calls $0.40.

River: ($2.31) 9:heart: (2 players)
BB checks, Hero ...

Pretty obvious you have the best hand here. He just tried to take a stab at it thinking your bet was only a continuation bet, it wasn't, and now he's checking the hand down.

I'd value bet about the size of the pot because it seems to me this guy might be dumb enough to call it even with something like second or third pair.

Well, he may have a weaker J in which case you can extract another bet from him, but I just don't feel TPTK is a good enough hand to bet him on the river. His min-raise on the turn is a sure sign of strength. I would check behind here and hope he doesn't turn over J4. Now if he had called the turn bet, I would bet the river without any second thoughts.

I agree the min raise on the turn is obviously no bluff and when people tend to raise me the minimum after i have been the aggressor on all the previous rounds of betting it tends to be a danger sign the fact that he checked the river might indicate he is trying to induce you to bet again.
I would also check the river and see what he shows, i also think he probably made two pair. If not and you beat him take note of how he played that hand poorly and use it to your advantage next time.

Also what do you do if he raises your bet on the river, not just the minimum this time but say 3/3.5 times your bet, do you feel you can call him with tptk???
Check is the best option here

thanks for the replies guys. RESULTS!

BB checks, Hero checks.

Final Pot: $2.31

Results below:
BB has 8d 7h (one pair, sevens).
Hero has Ah Jh (one pair, jacks).
Outcome: Hero wins $2.31.

People min-raise on 25 NL all the time for no reason what-so-ever. I think they usually trying to slow the raiser down so they can get a cheap showdown.

I put him on a jack w/ a bad kicker, but 2nd pair also makes perfect sense for the players at this level.