What is the connector on kinesis keyboard board called?
This is a JST SH Connector. JST is the "brand"/designer. 

Here's the digi-key part. https://www.digikey.com/product-detail/en/jst-sales-america-inc/B7B-XH-A(LF)(SN)/455-2252-ND/1651050 Please note that your connector may be a chinese variant or knockoff, but this shouldn't matter for you.

Here's how you can search for connector types: http://www.digikey.com/en/resources/connectors/rectangular-connectors