There is a built-in function in Tiny MCE for this, but configuring it into Umbraco differs a little from the Tiny MCE out-of-the-box configuration.
To configure these values into Tiny MCE in Umbraco you need to add a <config>
node to the <customConfig>
node in Config\tinyMce.config.
The following is the stock config, but formatted to successfully make it through umbraco's dynamic evaluation:
<config key="table_class_list">
[
{"title": "None", "value": ""},
{"title": "Dog", "value": "dog"},
{"title": "Cat", "value": "cat"}
]
</config>
This will configure the class options as per the TinyMCE documentation.