Hi JenJ - welcome to Rootschat!
Well, I have found Lucy's christening record and that of her brother. They're both on the International Genealogical Index (IGI) at
www.familysearch.org. You can see them there yourself...
Lucy Maria Illenden Bashford c. 20 Aug 1837 in Folkestone, Kent.
Fred Jn Benthan Bashford also c. 20 Aug 1837 Folkestone.
Fred's record also indicates that he was born 01 Jun 1829.
In both cases the mother is given as An Bashford (yes, with only 1 n - probably transcription error), no father given. I don't know what the Jn in Fred's is for (Jonathan?).
I tried looking for Illendens in Folkestone, but there don't seem to be many. All I could find was a William married in 1731, another William married in 1744 and an Edward married in 1731. Also an Eliz christened in 1664. All of these are listed as "Ellenden".
I couldn't find Ann's christening in Dover - I don't know how familiar you are with the IGI, but it's not complete. It has entries from A LOT of parish records, but there are many gaps in years and some parishes aren't represented at all.
I'm afraid that's all I can come up with right now... I hope that helps!
MJP