If the son served in the Mexican War, first check to see if there was a pension granted. If so, then the bounty land warrant application should be filed in with the pension file. If there was no pension, there will be a separate bounty land warrant application file. All of this would be at NARA, and these documents are not online or on microfilm as of now. I make color digital images, copying from the original paper at NARA.
Are you sure it was for 166 acres? That is a very unusual amount. It is probably 160 acres. You can search for the name at the BLM website at: www.glorecords.blm.gov
and click "land patents". Any land transaction listed there would have a small file of documents at NARA. You can look at the detail screen where it says, "authority" and if it says "scrip-warrant act of 1812" (or of 1847, 1850, or 1855), then that would be federal bounty land as a result of military service. You can click on the "document image" to see the patent certificate, which would include the regiment that the soldier was in. With that information, you can get the service record, pension file, and any bounty land application file and/or surrendered warrant file, all from NARA.