The easiest way is to trace backwards, starting at what you know. If Mum has her birth certificate that will give the names of her parents. By ordering the marriage certificate of these people you get their fathers names. Etc etc. When you get back to the early 1900's you can check on the 1901 census.
You can join ancestry.co.uk or findmypast.com to help you trace these certificates, though you will also be able to access this information free from your local library. Whilst in there get out a book on how to trace your family, it will have loads of tips in it.
There are also a number of monthly magazines devoted to this subject, have a look in Smiths.
Also join genesreunited. Someone may already have done the work. I have traced a first cousin that I haven't seen for 30 years on genesreunited, also more distant cousins, one of whom had traced one branch of my family back to the 1700s.