Generally, each team has one jungler, which means that there will be one lane of two, and two lanes of one. Since the middle lane will always be a solo lane, this means that top and bottom will have varying numbers of champions, depending on the lane.
Top is usually considered a solo lane because the bottom lane is closer to Dragon, an important mid-game objective. Dragon is hard to solo for most non-junglers, and by keeping 2 champions in the bottom lane, it becomes quite viable to 3-man Dragon after a successful gank (or simply a double-kill). While top lane is closer to Baron Nashor, it takes quite a bit more effort to kill him - more than 3 champions in the early game can probably afford to put out. By the time Baron becomes available, the game is usually nearing the end of the laning phase, after which the difference between top and bottom lane is moot (because all 5 champions are roaming).
With a top solo lane, the champion has a leveling advantage and the potential to farm more minions (simply because there is no allied champion to compete with). Additionally, one must consider that if the other team does not have a jungler, they need to be able to successfully last 2 vs. 1. Most often, this means champions with good sustainability are better suited to the lane, such as Cho'goth, Mordekaiser, Vladimir, or Nasus.
Conversely, there are some champions who do not do well in a solo lane - these tend to be support champions like Soraka, Sona, or Janna who perform much better with a partner to buff/shield/heal. These champions usually go to the bottom lane to ensure that they are in a 2 vs. 2 situation.
In instances where there is no jungler on either team, both top and bottom will have an even number of champions. In this case you may be asked to "switch lanes" not because one lane is "better suited" than the other, but because switching lanes means changing which enemy champions you're fighting (and presumably, have been dying to).
Best Answer
A major difference between 5s and 3s is how much lane control you need to win the game. To summarize: for 3s you want high dps champions with a lesser focus on team value, and in 5s you want high lane control champions with great team value.
In 5s you value champions that can hold their own lane well, because teammates will leave their lane open for minutes while they travel to help you. In 3s you value champions that can help other lanes and contribute lots to ganking, because traveling lane to lane takes very little time.
Champions like Heimerdinger and Soraka do well in 5s because they both have great lane control. Heimerdinger's turrets and passive heal allows him to push a lane very well and stay in good shape, meaning he doesn't need to leave the lane for a heal and spend a minute running back out. Soraka can heal herself and replenish mana, so she almost never needs to leave a lane.
However, in 3s they are less handy because they are squishy and susceptible to being ganked. They also provide little value when it comes to ganking in other lanes because they are not damage-per-second champions. To contrast Heimerdinger and Soraka, Tryndamere is a champion who shines in 3s. He can quickly travel from one lane to another by spinning over walls, and his dps makes ganks easy.
Another thing to note is that you require team composition in 5s, as any one champion can be focused down by 5 others at a time without a team backing them up. In 3s it's less of a problem because champions last longer in team fights as less damage is going around. Tryndamere in 5s can be crowd controlled by 5 other champions and he provides no aid to his teammates. However, in 3s he can solo two champions and likely win.