My mom in law is moving from Bulgaria to US and needs to get a large sum of money to the US from Bulgaria for living expenses. What is the most cost effective way to transfer the money from Bulgaria to US? If she transfers to my bank account will I have to pay US taxes on the money if I give back to her upon arrival in the united states?

Better to have her open an account in her name with an international bank, deposit the funds there and then she would be able to access the funds when she gets here. Wells Fargo, US Bank, Chase and Citibank can do those services. IRS frowns on straw purchases.