You don’t say where your father’s home is located, and the relevant laws vary from jurisdiction to jurisdiction. So the following answer should be viewed as general information only. At some point you might want to consider speaking to a lawyer or solicitor.
If I may, let me address a minor point of terminology first. In virtually all jurisdictions, a bank’s mortgage is only a lien on the ownership of the property. Nevertheless, for historical reasons, some folks refer to that as, “the bank owning the house” and paying off the mortgage as, "buying back the house." Technically, in most jurisdictions, that’s wrong; the bank only has a lien on the ownership. I believe that is why the bank told you that you had to “buy the house”; I believe that all the bank meant was that you would have to pay off the loan and replace it with a new one. That is what is called a “refinance”. I don’t think that you have to be concerned about paying the purchase price or value of the home. You might want to check again with the bank.
If you weren’t contemplating a construction loan, as long as the bank continued to accept your loan payments, whether or not to pay off the loan would depend on how favorable your father’s loan is to you. But, if you want to get a construction loan secured by a mortgage, it is likely that you will have to pay off your father’s mortgage at the same time. Why? I will try to explain.
There can be more than one mortgage encumbering a home at the same time. But when there is more than one mortgage, they are not equal in priority; the so-called “first mortgage” is “prior to” the “second mortgage” and, if there is a “third mortgage” it is "junior to" the “second mortgage”, etc. This is critical because, if they are not paid, and the first mortgage forecloses, it likely “wipes out” or destroys all of the subsequent mortgages and those junior lenders are left with no mortgage at all.
You can see why a bank is at a disadvantage if it holds a second mortgage. Although not terribly uncommon, having more than one mortgage on a home at the same time is in the minority because most “institutional lenders” (banks, insurance companies, credit unions etc.) are not willing to put themselves into the more dangerous “second mortgage” situation. For that reason, generally speaking, a new institutional “construction lender” would not be willing to put itself into a “second position” and it is unlikely that your father’s lender would either. (There are some ways to get a construction loan with a second priority mortgage but those can be more expensive, and the chance that your lender fails to have the construction funds when you need them, may be increased.)
All this means is that, to get a construction loan from an institutional lender, you will probably have to use the new construction loan also to pay off your father’s mortgage and bundle the loans all up into one mortgage. This process would “refinance” your father’s mortgage for $100,000 and finance the construction for $100,000, resulting in a new loan for $200,000 and a new mortgage for $200,000. Depending on the rates being charged for your father’s old mortgage and the rates that would be charged for your new mortgage, that may or may not be financially beneficial.