Hojicha is a green tea which is made from bancha, a low grade green tea, and cooked slightly; this very inexpensive green tea often comes out brown because it is discolored by oxidation. Other than this variety, and some very stale bancha, I can't think of a Japanese green tea that comes out brown. Some stale kukicha might come out brown, and low quality genmaicha made with poor quality kukicha could be brownish from the combination of excess oxidation and the toasted rice.
Most of the non-Japanese green teas I've seen sold as "green tea" and some lower quality teabags marked as "sencha" are broken down tea leaves that can easily oxidize in the packaging to the point where they are, at best, yellowish.
However, if you use even a moderate quality sencha, and it isn't stale, it should come out green. Most good quality kukicha and genmaicha will at least come out greenish-yellow. Since most Japanese restaurants use a fairly inexpensive second-harvest sencha, I imagine you're either buying hojicha or simply using a very stale, low-quality sencha.
Chinese green teas are sometimes slightly yellow, but if they're actually yielding a brown brew, they, too are either stale or are simply mislabeled oolong.
I used to sell a fair amount of green tea when I ran a small scale import business. I do recall even matcha products turning brown after being exposed to excessive sunlight; one of the bakeries I know was constantly running into issues with their matcha pound cake slices on sunny days, especially those pieces in the front of their pastry case. So it's reasonably possible you simply have tea which has been stored improperly; you're best off with a nitrogen flushed tea and it should be used within about 6 months of opening the sealed package. If your tea comes unsealed, it probably isn't in very good shape to begin with.
In any event, my recommendation is to try a first-harvest high-mountain grown sencha that is sold in nitrogen-flushed aluminum packages. Depending on the style, the brewed color will be either very green or slightly yellowish if brewed in the typical way. You can either do a very brief, several second infusion at near-boiling, or a longer infusion at around 80 celsius. I like it both ways.
Gyokuro, if you can afford it, is extraordinarily green, but is unlikely to be served at a restaurant without an additional charge (it's essentially the same leaves used to make matcha, except rolled instead of being milled).
There are couple of factors at work when re-steeping green tea: temperature, time, and the quality of the tea (the size & way it's been processed). Green tea is supposed to be brewed with water that's been brought to boil and allowed to cool to 167 - 176 degrees Fahrenheit (though many people simply heat the water to that point or what they eyeball as hot but not boiling). Green tea is typically steeped for 1 - 2 minutes. And green tea, especially loose leaf, but also with higher quality tea bags, has furled and sometimes rolled leaves. The lower temperature, the short steeping time, and the curled leaves mean that flavor is still left within the leaves even after multiple steepings. The leaves continue to expand and release in subsequent baths. I don't, unfortunately, know any of the exact science--what's being released when, etc.
However, a possible alternative to boiling & brewing individual teas when done or simply using more tea (the typical choice when brewing larger quantities), you can bring your water to temperature, steep, pour it into a preheated or heat retaining container, and repeat the cycle until you've brewed your multiple cups in one go. As far as I know the tea leaves don't need to 'rest' in between brewings.
The reason green tea can be re-steeped is also the reason most black teas can't. Black tea is subject to boiling water for a longer period of time (typically Western use is 3 - 5 minutes; some cultures favor an even longer steep for the deeply bitter tannins). The hotter and longer steep pulls more flavor out of the leaves faster. Some black teas, especially high quality whole-leaf, can be re-steeped, however. Most bagged black teas--and a lot of loose leaf--can't.
The heat & steep time is dictated by the processing of the tea once it's picked. Green tea has minimal processing--pluck, wilt, shape & dry. Black tea is allowed complete oxidation (called fermentation). Fermentation breaks down chlorophyll, releases tannins, and forms many of the taste & aroma compounds that typify black tea. Oolongs and other varieties are subjected to varying degrees of fermentation. The size & shaping of the tea leaf--allowed to remain whole, broken, rolled, or (for most bagged tea) cut into fannings and dust, also affects how tea should be initially steeped and whether it can be re-steeped. Highly rolled oolongs, for example, often hold up to more steepings than green tea and are even said to taste best only on the 3rd or further steeping.
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I know that with pu'er tea (not a green tea), they rinse and 'awaken' the tea with the hot water first - dumping out the first brew (seemed like a waste to me). They say the 2nd is good, but the 3rd cup and on is much better ! I never noticed the color of my green tea changing, but I must admit i like the second pot better than the first - when re-brewing same tea leaves. perhaps the oxidation from the leaves is brewed of in the first pot?