The busy shopping street leading from the Kaminarimon gate to the temple is the covered Nakamise (仲見世) arcade, selling all sorts of Buddhist paraphernalia as well as assorted tourist kitsch.
This is one of the best places in Tokyo to buy souvenirs (the other being the Oriental Bazaar in Omotesando), but note more expensive items such as swords and kimonos are likely to be of inferior quality. Slightly nicer crafts, rather than mass-produced kitsch, can be found at good prices if you walk up to the temple, turn right, and turn right again on the first small street running parallel to Nakamise. You will see plenty of small shops in this general area which have better quality souvenirs and gifts, like handkerchiefs, strings of hand-made silk balls, hairclips, etc.
A more offbeat shopping option is Kappabashi-dori (かっぱ橋), best reached from Tawaramachi station on the Ginza line. This is Tokyo's restaurant wholesale district, which sells plastic food, metal spatulas, deep fryers and an immense variety of affordable crockery. Some shops sell only in wholesale quantities, but many are happy to sell single items and factory-made Japanese pottery (which to the casual eye is indistinguishable from the handmade kind) can sell for as little as ¥100 a piece. Another bargain is high-quality Japanese kitchen knives, which are generally much cheaper in Japan. Note that most stores here are closed on Sundays.
Asakusabashi (浅草橋), two stops south on the Toei Asakusa Line, is a wholesale district which these days is known for its shops specializing in bead craft supplies. There are also many stores selling traditional Japanese festival and party supplies.
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