Bound and Gagged: The Villains Of The Week in this show employ this relatively often, one example being in "Maximum Insecurity" when Jasper is tied to a chair and gagged with his own sock while being held hostage.
The Cheerleader: Subverted when the 'dumb cheerleader' turned out to be incredibly smart, and was dosing the team with the 'fountain of youth' in secret. The reason she pretended to be dumb was so boys would like her.
The show seems to find any excuse it can to get Henry shirtless. Even when he's not shirtless, we still get plenty of glimpses of tummy due to the fact that pretty much every acrobatic move Henry does culminates in his shirt riding up - usually quite a bit higher than the move would normally result in, but hey, I don't hear anyone complaining.
Henry and Jasper in suits.
Female Gaze: After it's mentioned in a roundabout way that Henry goes commando ("It's a formal dance Henry, so wear underwear!"), the camera follows his bum as it walks upstairs for a little longer than necessary.
Fish out of Water: Henry lived in pretty much everywhere on Earth except in urban society.
Meaningful Name: An Adventurer Archaeologist named Henry. Where have I heard of this before... Also, Smithson High is named after the founder of the Smithsonian Institution, which, as every good Indy fanboy knows, keeps the dangerous magical artifacts out of the hands of the crazies. In real life, the Smithsonian is simply the largest museum in America; pious skeptic Ales Hrdlicka's been dead longer than most people here have been alive.
There's Sven Erickson who studied the impact of vikings on American history. Considering the most well known viking is Leif Ericson...
Mighty Whitey: Henry has apparently lived with members of every African and Asian culture and excelled at all their skills, though we don't know how good he was in comparison to the natives. Of course, this does leave him with problems being a standard North American teenage boy. It's not clear if this is supposed to be a Deconstructed Trope or not.
No MacGuffin, No Winner: Henry destroys a rare plant so no one will be able to abuse the powerful stimulant it produces.
Actually, he saves one of the seeds for someone responsible enough to study it.
Nobody Here But Us Birds: Played straight to lure the bad guys around the museum in "Maximum Insecurity". Well, they thought it would work, till Jasper fails to do so. Played straight later in the climax.
Red Oni, Blue Oni: Henry is obsessed with figuring things out and steadfastly refusing to be tied down by convention, while Jasper is preoccupied with fitting in and doing what his father tells him (for the things his father pays attention to, at least).
Sailor Earth: When they're not writing Jasper/Henry slash, fanfic authors enjoy pairing up the already hinted Maggie/Jasper relationship, and writing in an OC (usually very well-travelled as well) to act as a Love Interest to Henry. These can stray into Mary Sue territory.
Screwed By Being On The Network: It's not a bad show, and if it were on almost any other network, it could be well-received. But because it's airing on Cartoon Network, no matter how much they promote the show a lot of viewers aren't willing to give it a chance.
To Be Continued: Strangely, the season finale ended with a (literal) stamp of this, even though the plot of the episode wrapped up. It's questionable if the second season would actually start with the three of them solving some mystery in the Gobi.
The Unreveal: "Maximum Insecurity" started off with Jasper and Henry making a deal that whoever got the better exam scores would plan their weekend. After an offer from Mr. Bartlett to excuse them from taking the test so they could fit in both plans, Henry refuses because he'd already studied so hard. We never see how the boys do. Most assume Henrygot the better score.