Yes it does make miniaturised hairs grow in thicker, and thicker hairs are capable of growing longer.
I'm just going to take a guess here regarding the mechanism
Rogaine increases the flow of blood to the scalp right? So I assume that the blood flow carries amino acids, the building blocks of proteins (keratin) to the hair follicles so the hairs produced have more keratin to work with and become thicker. And thicker hairs are stronger so can grow longer, just like how a thicker plant stem can grow longer without breaking under it's own weight.
If that is the case, then I wonder if it just causes hairs top regrow at full thickness, or if it can induce follicles to produce multiple hair strands?