First let's get some terminology clear. This:
<Lifestage>Adult</Lifestage>
is an element. It has two tags, a start tag and an end tag. It has an element name (or element type name) of "Lifestage", and a value of "Adult". "Adult" is not the value of either of the two tags. This may all seem pedantic, but it's a lot easier to avoid misunderstandings if people use the correct terminology.
It's not clear what your processing context is, but assuming that your context node is the element that is the parent of the MEMBER_DETAILS elements, then I think you want something like
select="MEMBER_DETAILS[Lifestage='Adult']/FirstName"
With your sample data, this path expression will return a node-set containing two elements.
Michael Kay
http://www.saxonica.com/
Author, XSLT Programmer's Reference and XPath 2.0 Programmer's Reference