Evidence is going to be the key to confirming the family story so you are probably going to have to research your own ancestry more completely to get the evidence for the assignment.
I don't want to put you off but tracking back to the relationship youur mother mentioned will be quite a feat to achieve (see below) - it may be best to simply research your own ancestry and see who or what you discover.
Michael's suggestion of FamilySearch is a huge free resource but also most local libraries around the world have a membership of http://www.ancestry.com
which will give you access to a large volume of additional data, and vast numbers of smaller websites are linked from a huge collection of genealogical resources at http://www.cyndislist.com/
Additionally you may find that Google
searches can help you - check out the Advanced Search options to combine strings in different ways to find web pages from other people's research. As you traverse your ancestral tree you double, triple or even quadruple the number of relatives with every generation you go back and one of these may have published something online about your common ancestors. Try searching for the earliest couples that you are aware of with their marriage year like this:
"Mfirstname Mlastname"|"Mlastname, Mfirstname" "Ffirstname Flastname"|"Flastname, Ffirstname" 1867|1868|1869|1870|1871
This effectively means, search for a webpage that contains names for both ancestors written in either order that also contains one of these five years.
Confirming your mum's statement...
If your mother's GGGGmother was 3rd cousin to Winston Churchill's mother then that would make the common ancestor you need to track down your GGGGGGGGparents (a 1st cousin is someone who has a Grandparent in common - one G). Based on your mother's statement (and assuming that each generation is born an average of 25 years earlier than the following) you need to track back around 250 years from your own birth year. This simple maths is confirmed by Jennie Churchill's birth year being 1854 - her distance from your common ancestor 4 generations (3rd cousin) before her own birth is approx 100 years.
Good Luck !