Deliberate mistake: The police car Steve & Danny are driving on the bridge as they approach the freighter has 467 on the roof, which then becomes flipped & backwards a few shots later as they take a sharp right, only to become 467 again (as Danny has conversation with Rachel). (00:38:10 - 00:39:05)
Factual error: The aircraft carrier towards the beginning is captioned as being the USS Enterprise (CVN-65), where Lt Rollins is stationed. It is obviously not the 'Big E', but a much smaller escort carrier or amphibious assault vessel. Notable differences are not only the sheer size difference, but also lacks the angled flight deck with the protruding catapult brakes, the second port side aircraft elevator and both the superstructure and bow shape are wrong.
Continuity mistake: In the scene where McGarrett's dad's house is seen for the first time from ocean-side, deck on the top floor has furniture (wicker two-seat bench and two chairs under double windows with long table to the right) and the lower level has no furniture in grass to left of the ramp, and two large vases on either side of the ramp. Then two shots later (same scene) as Steve approaches under the crime scene tape, there is no furniture on the upper deck but now furniture to left of ramp and vases are missing. (00:10:25)
Continuity mistake: In the long bickering with Steve while they kinda talk with Sofia, Danny picks up the badge the Steve put on the table, put it down before the cut talking about bad relationship decisions, but in the next shot he's gesturing again with the badge. (00:23:10)
Continuity mistake: Steve and Danny sit at the diner's where they need to meet with the fugitive's girlfriend. As Danny gives Steve the speech about the "mammal to mammal" interactions he puts his toothpick in his mouth, but it vanishes in the front view, only to be plucked back at the beginning of the next shot. (00:22:35)
Continuity mistake: Inside Steve's office, Mary says that she would have taken anything from their dad, even disappointment,"because at least it would have been something." In that close-up, her hair is noticeably different than in the rest of the scene, much thinner over the shoulders. (00:19:45)
Other mistake: The episode makes sure to mention the real places where the action happens. The fugitive escapes from Halawa Correctional Facility, and dumps the car at Kakaʻako Beach Park, which is about 25 minutes away. Danny and the others arrive at the location way before Steve, who was at the airport, which is roughly 15 mins away from the penitentiary (kinda odd that they'd happen to be even closer than him, but fine). They have then the footage to investigate, the guard to talk to, the game of hoops with the prisoners, go back to their precinct, but despite the huge headstart Ellers had, they catch the car that was abandoned "15 minutes earlier." Moreover, Chin calls Danny to tell him about the car only after he already investigated the location and interrogated the witness. It can't even be 15 minutes, because 15 minutes is when the fugitive bought the products from the market, and he still needed to dye his hair.
Continuity mistake: Kono and Danny do a bit of theatrics distracting Frank Salvo and his men that are about to waste cousin Sid. The henchman behind the Samoan gang member is not touching him in the wide shot, but has both hands on his shoulders in the closer cut. (00:34:15)