What does 'a slither of' mean?

Solution 1:

This is a misspelling of sliver

any small, narrow piece or portion

So a slither of lemon tart or apple cake really means a sliver of lemon tart or apple cake. It is implying a thin slice of the cake or pie.

Apparently this mistake has made it into common parlance, as attested by this entry from oxforddictionaries.com

Slither noun - a sliver

As we can see from the origin of slither

O.E. slidrian "to slide on a loose slope," a frequentative form of slidan "to slide" (see slide). Meaning "to walk in a sliding manner" is attested from 1848. In ref. to reptile motion, attested from 1839. Related: Slithered; slithering.

when compared to the origin of sliver

late 14c., from obsolete verb sliven "to split, cleave," from O.E. toslifan "to split, cleave," from P.Gmc. *slifanan.

They are not related. It isn't until the beginning of the 1900s that slither appears where sliver should, in writing.