There's a few game hubs in Tokyo, but Akihabara is the classic techy gamer paradise. Plenty of new and used game stores, arcades and curious little shops. In particular:
Super Potato, on the upper floors of a back alley building off the main strip. A veritable treasure trove of vintage video games (Nintendo, Sega, Sony plus weird and wonderful platforms you've never heard of). Lots of rare machines and games too (including a working Virtual Boy). See their website for more info.
The Sega Arcades (8 floor one in Akiba) - 8 floors of UFO catchers, fighting machines, shooters, the works). Take the Electric Town exit from JR Akihabara station and you can't miss it
The TRADER chain of shops - 5 in all in Akiba, with lots of rare / used games going back to PS1 days. TRADER 4 is probably the best one, but they're all full of interesting odds and sods.
Yodobashi Akiba - a mammoth electronics department store with a massive floor dedicated to recent platforms / hardware / software. A giant version of your usual Gamestops / GAMEs. It's on the other side of Akihabara station from Electric Town, and is so large you'd struggle to miss it.
For more stores see this great Guardian article.
If you are over 20 there are also lots of cool video game bars to try out. There's a good few in this GamesRadar article.
Of these, I've visited 8 Bit Cafe in Shinjuku - suitable for nerds and non-nerds alike, with a comically large cocktail menu, each entry based on some video game (the Dr Mario one in a beaker with gel pills is excellent). See their website (in Japanese) for more details. They're based at Shinjuku 3-8-9, Shinjuku Q Building 5F (it's a bit of a nightmare to find, but the linked article has some good images to find the exact stairwell to traverse).
I should add, for the bars, some of them are video game themed, but not necessarily what we would consider 'gaming cafes' (i.e. you can drink but it's not a place for playing games).
If you fancy a futuristic theme park experience, try the incredible Joyopolis in Odaiba. A massive indoor theme park made by Sega with incredible rides (real rally cars playing Sega Rally, a House of the Dead roller coaster), a big arcade, Sega-franchise exclusive arcade machines, and a general all-round "this is insane" vibe. See http://tokyo-joypolis.com/language/english/. It's at DECKS Tokyo Beach, 1−6−1 Odaiba, Minato, postcode 135-0091.
Of course, if you're a real pilgrim you could get pictures outside the Square Enix office in east Shinjuku and the Nintendo office in Asakusabashi (5-21-5 Asakusabashi, Taito-ku, Tokyo 111-0053).
If you've anything specific in mind though leave a note in the comments and I can have a think.
Update - If you have an international drivers permit you can also do real life Mario Kart around the streets of Tokyo, which is probably about as gamer-cool as it gets. Find out more at the booking page
You might also be interested in taking a tour of the Sony Building in Ginza - it's more than just games, but they usually have a cool selection of the latest Playstation games, gadgets and gizmos, as well as a mini-cinema that sometimes features games / trailer. Take a look at the Sony Building English website.