Someone with Bipolar disorder may be medically eligible for SSDI (or SSI). It depends upon the severity of the condition. Also there are non-medical requirements for these programs that must be met as well.
Again, regarding a child with ADHD, the child might be eligible for benefits under the SSI program. The requirements can be found in volume 20 of the Code of Federal Regulations, Section 416 et seq. Subpart P contains the medical requirements and 416 in general sets out the non-medical requirements.
I would urge you to obtain the services of a competent attorney who specializes in SSDI/SSI cases. This is a complex area of law that can be very diffcult, even for experienced practitioners. No fee can be charged unless and until your case is won and the fees are strictly regulated by SSA (up to 25% of the back benefits or $5,300.00, whichever is less).