My nearly 16 yr old son was diagnosed with breast buds with they go away and when (he's 5'4 3/4, 125-130 lb?

Male breast buds: unlikely to be breast buds unless he is taking hormones, more likely to be male adolescent gynecomastia which will resolve on its own.

Yes: They should go away. Give him more time in puberty phase. This is normal.

In time: Pubertal gynecomastia will occur in 1/3 to 1/2 of males & generally last about a year before fading.Up to 3 years is noted occasionally before disappearing. Overweight kids can appear to have it from just fat accumulation.If present after 3 years, surgical removal of the glandular tissue & liposuction of any associated fatty tissue will do the job.