Dependency would form the bases for the extension of an "O" visa.  If you have a Thai child or children you may stay in Thailand based on dependancy. You would apply for an "O" visa based on having a minor child in Thailand. There are certain regulations governing this stay. You would apply for an "O" visa a Thai embassy abroad which is valid for 90 days and extend it in country thereafter.

Requirements:

1. Application Form (T.M. 7)
2. A copy of your passport or document used in lieu of your passport
3. The applicant’s 4x6 cm. photo.
4. Marriage certificate.
5. Extension fee, Baht 1,900.
6. Evidence to identify that the supporter or the one being supported has Thai nationality or is residing in Thailand.
7. Evidence to identify that the supporter or the one being supported has Thai nationality or residence in Thailand.
8. If the applicant is the supporter, he must submit financial evidence, proof of employment, work permit and tax receipts

Should you not be able to manage the process yourself, contact a lawyer in Bangkok to manage the process for you.