Can my share buttons include the product name in the shared message?

Yes. In the Share URL you can use two placeholders:

• %%URL%%: replaced with the product page link
• %%TEXT%%: replaced with the product name

To change a button, go to Marketing → Social Sharing, click Edit on it and add %%TEXT%% where the name should go in the Share URL. For WhatsApp, put %%TEXT%% and %%URL%% together in the text part of the address. Click Update, then test the button on a product page.