Whilst overdrives aren’t there to completely change your sound, many do add some tonal qualities that players love, so the inherent sound of it should be considered. Others really let the sound of your guitar and amp shine, so if you’re happy with your tone as it is, but want a bit of extra attitude, then you’ll want to look for an overdrive pedal that’s more transparent.
Some feature more tone-shaping controls than others too. If you have a specific sound in mind, then look for something with more EQ controls, as these will give you more ability to fine-tune your sound. You can also use these to combat troublesome frequencies within your amp, for example, many metal players use a Tube Screamer, or something similar, to push the front end of an already distorted amp a little more, and to tighten up the bottom end. If you’re using an overdrive pedal for solos, then dialling in more high-mids and top-end can help you cut through a live mix better.