here's what the IRS says.
https://www.irs.gov/businesses/small-businesses-self-employed/do-you-need-a-new-ein
to summarize a single-member LLC is treated as a disregarded entity - ie a sole proprietorship. so as long as they remain single-member you do not need a new EIN if you change the name of the business. as a matter of fact if you operate 20 single-member LLCs each performing different activities so separate schedule Cs need to be filed you use the same EIN for all 20.
