I assume you know that the group and raid windows already provides this functionality. Clicking a player's health/power bars target's them. Custom group windows often lay script buttons over other elements in the window but rarely cover the health/power bars.
Assuming you still want to build this as a separate mod, try the following
Code:
onPress="
target=Parent.Parent.GroupMembers.GroupMember1.MemberInfoPage.MemberInfo.Name.LocalText
target=Parent.Parent.GroupMembers.GroupMember1.MemberInfoPage.MemberInfo.Name.LocalText
"
This assumes the script is applied to a button at the root level of a custom window.
Note the command is issued twice to ensure it works on a single button press. Slash commands applied via script like this often don't work when first used, so need to be repeated, either in script or by pressing the button again.
There is almost certainly an alternate method using DynamicData for group member names, I'm just to lazy/tired to dig out that code.