How do I change the "Standard Shipping" name customers see at checkout?

Go to Settings → Store Settings → Shipping Setup. In the Manual Shipping section, type the name you want into Label, for example "Home Delivery", and click Save Changes.

That label is shown next to the shipping charge in the cart and checkout summary whenever your manual shipping applies.

If you use the Shipping Zone plugin, customers whose address matches a zone see the Label of that zone's methods instead. Edit those under Plugins → Shipping Zone → Edit on the zone, then click Save zone.