If that picture is correct, it's just a bog standard CR2032 with a connector on the end. Given that the connector part will already be in your mate's computer, you should be able to open up the taped end and just insert a CR2032. Almost every discount store sells them cheaply, e.g. https://www.bmstores.co.uk/products/panasonic-coin-battery-6pk-292765
Just a quick one Knocka…...The socket attached to the battery may be the fly in the ointment, and I cannot tell whether there is a built in choke or small resister on the "Acer" supplied part. However you can use the old connectors to hook up a new battery. Looks like a mare on the underside of the mother board. Good luck.