One way to look up maiden names for free is to access a person's profile on her social media pages. Many pages have a section that identifies a woman by both her married name and her maiden name. Another option is to consult the genealogy section of a local library for public records on the person.
It is also possible to utilize social media is to find the person's father or siblings through the person's profile, who are usually identified as such in the person's "friends" or "about me" sections. A father or a brother's last name is likely to be the person's maiden name as well. Search the library for access to marriage certificates, birth and death records, military records and wills. If there is any cost in this tactic, it is only for printing copies of records if you choose to do so.
If the person whose maiden name you are searching for is your own mother or the mother of someone you know, look at your own birth certificates, which typically contain the mother's maiden name. Other options include finding and examining the marriage certificate or death record of the person. If these are not available, interview family or friends or find old letters or address changes.