See the Note for her son in law William de Samlesbury to show why her daughter Avina was not heir.  At the end I put:

The VCH Lancs entries for Brightmeit and Barton, the properties of her mother Cecily, who was indeed sole heir of them, these entries make it clear that Cecily had a son Gilbert who was heir of her mother who was his grandmother, Edith de Barton, which Gilbert had a son John and whose male-line offspring survived in that area for several generations.  The confusion by the VCH writer of the Samlesbury article may have been caused by the fact that Gilbert, on being declared heir of Barton, fairly promptly assumed his grandmother’s surname and became Gilbert de Barton, likewise his offspring.  This left Avina as the only child of Cecily who had probably retained, until marriage, the surname of Notton, thus leading to the assumption that she was an heiress.  Pity, as it had seemed that there were several quarters to be had.
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