There are two approaches explained in the great tutorial be Alex Lockwood: http://www.androiddesignpatterns.com/2016/08/coloring-buttons-with-themeoverlays-background-tints.html:
Approach #1: Modifying the button’s background color w/ a ThemeOverlay
<!-- res/values/themes.xml -->
<style name="RedButtonLightTheme" parent="ThemeOverlay.AppCompat.Light">
<item name="colorAccent">@color/googred500</item>
</style>
<Button
style="@style/Widget.AppCompat.Button.Colored"
android:layout_width="wrap_content"
android:layout_height="wrap_content"
android:theme="@style/RedButtonLightTheme"/>
Approach #2: Setting the AppCompatButton’s background tint
<!-- res/color/btn_colored_background_tint.xml -->
<selector xmlns:android="http://schemas.android.com/apk/res/android">
<!-- Disabled state. -->
<item android:state_enabled="false"
android:color="?attr/colorButtonNormal"
android:alpha="?android:attr/disabledAlpha"/>
<!-- Enabled state. -->
<item android:color="?attr/colorAccent"/>
</selector>
<android.support.v7.widget.AppCompatButton
android:layout_width="wrap_content"
android:layout_height="wrap_content"
app:backgroundTint="@color/btn_colored_background_tint"/>