I am presuming here that the child in question is one who was not formally adopted, and was kept within the extended family (not put into any sort of institution/orphanage/etc), but not raised by his/her birth mother.
Records relating to the birth
A birth certificate will normally make it clear if the birth was out of wedlock, but may also contain further useful details. For Illinois, statewide registration began in 1916 (general compliance by 1922), and a birth certificate after these dates (towards the end of your stated period) will list mother's birthplace and age, which even if a little inaccurate may allow you to distinguish between candidates.
If no other information other than mother's name is listed, the next most important field is the address of birth. This may turn out to be a hospital or mother's home which holds its own records of birth or admission (these may be difficult to access, however). Or, if the birth was at home, you may be able to link this address to other members of the family who lived there.
Depending on the religious faith of the family, there may also be a baptism record in addition to (or instead of) a birth certificate. If you are uncertain of precisely where the family lived or which church they attended, this may be difficult to track down, however. This may give slightly different information than the birth certificate, or useful additional information (i.e. names of godparents).
Records after the birth
Marriage and death records for the child may shed further light on parentage, but can be very hit and miss as often the child themselves was not fully aware of the details. Any details on these records should be taken with a pinch of salt - particularly details of "father" who was sometimes an invented character!
A child born 1900-1930 in the US who survived to adulthood should appear on at least one publicly available US census record (the latest available being 1940 at the time of writing), which can be a very useful way of establishing which extended family group the child belonged to.
Listed relationship to relatives in the census may be vague ("cousin") or not be entirely truthful - i.e. the child of a 14 year old unmarried mother may appear in censuses as her "younger sibling" - raised as the child of the grandparents. A large gap between children or a "mother" who would have been well over childbearing age can be signs of this. Again, in this case where you know the listed mother in the census was not the birth mother, it becomes almost entirely certain that the real mother is a part of the same family.
Another place where relationships between family are sometimes clarified are in documents surrounding death - not the death records specifically but in obituaries, death notices, and probate records. Money may be left to a "grandchild" or "nephew/niece", or they may be included in lists of surviving relatives.
If newspaper records for the town in which the family lived are indexed, these can be a goldmine - especially for small towns. Expect not to see specifics of the family dirty laundry, but general notices mentioning visiting family ("John Smith of Chicago is visiting his grandparents, Mr and Mrs WT Smith, in Smalltown"), mentions of relationships in court cases ("John Smith, aged 8, nephew of the above witness..."), etc. If the names are very common in the area these references might be hard to disentangle.
