In doing animation for abstracted things like logos, you're really asking yourself some of the same questions as when you're designing the logos themselves. You're representing some entity, or some set of ideas, which may be very broad or nebulous, so you have some strategies that help you communicate something simpler, or more personable or relateable about that nebulous thing. Maybe the logo design makes a reference to an external idea that seems relevant, or maybe it's a metaphor for an ideology, or maybe it's a pastiche, or a simple pun, or an anthropomorphic character, etc. Whatever that logo design strategy is, the end result is a single, contained image which tells the viewer something relevant.
The animation of that logo can really follow a similar design process, where one of many strategies can be employed. Say the logo is for a rock climbing gym, and is an illustration of a small rock. The design isn't a literal strategy, because a small rock doesn't represent a rock wall, or climbing, or gyms. It's a slight pun (a stupid pun, granted, gimme a break). The question you ask now is: what can I communicate about this climbing gym by using animation, with this rock logo as a main element? And those ideas that you want to communicate, typically in support of or in juxtaposition with those inherent in the logo, will tend to suggest different strategies. And the balancing act you'll play will be to do it in a way that is still fitting for both the logo and the gym itself. So you might make a reference to climbing by animating a bunch of small rocks falling, as if down a steep surface, with the hero rock coming to rest in its place, all in a way that implies that someone's foot just loosed gravel, or that there's a micro avalanche on a rock face. That idea is contingent on your ability to animate finely enough that you can convey the specificity of the situation, but if successful, takes the visual pun of the logo and adds a new spin to it, with new meaning.
It's important to note that some strategies are going to require more or less work than others, but that once you have a strategy or solution in mind, you're already in a better position than if you were just winging it with some camera moves and a combo of wipes and slides and other arbitrary bullshit. When you build arbitrary motion, most importantly you're not really doing anything for your viewer, but you're also not helping yourself much because it's hard to tell whether your design is succeeding if you don't have a goal against which to measure it. So find a motivation for your animation by looking at it as a design problem, and accomplish the goal you set by looking at what the animation is actually communicating. If the motion isn't conveying much, it probably isn't motivated, and you just need to dig into the problem like a designer until your goals are really clear to you.