Here's a simple version that just shows the phone number:

View phone number for family member: <$list filter={{!!family}}><$radio tiddler=<<qualify "$:/temp/FamilyContact">> value=<<currentTiddler>>>&nbsp;<<currentTiddler>></$radio>&ensp;</$list>

<$let stateTiddlerName=<<qualify "$:/temp/FamilyContact">>>
  <$transclude tiddler={{{ [<stateTiddlerName>get[text]] }}} field="phone"/>
</$let>

Notice that we don't transclude the tiddler $:/temp/FamilyContact itself, like <$transclude tiddler="$:/temp/FamilyContact"/>, we transclude the transcluded value of $:/temp/FamilyContact (using the filter [<stateTiddlerName>get[text]]). You might have to think about this for a moment: we stored the name of the selected contact in the tiddler $:/temp/FamilyContact, so we don't want to transclude the tiddler $:/temp/FamilyContact – we would be trying to get the value of the phone field of $:/temp/FamilyContact itself (which is presumably empty). Instead, we want to transclude the phone field of the tiddler which is named within that tiddler (i.e., the contact tiddler we selected). This means we effectively have to transclude twice.