From a marketing perspective I've been involved in a couple rebrands / renaming. There is a very high probability of a 3 to 6 months drop in website traffic combined with a drop inbound leads. This drop can be significantly longer if you don't correctly handle website migration correctly. So I would not recommend a name change when taking over a business. Much better to focus on understanding and improving the business vs a name change which will likely create no short-term value.

Depending on the kind of business, I am in professional service business. I planed a rebrand 6 mo after after acquisition, and 6 mo execution. we first communicated to all our customer right as we made rebranding decision, then we ran both name (logo) on website, but kept old domain, we start doing new communications like invoices in new name, and after about 6 mo we removed old name on the website and redirect old domain to new. With that there was really no issues for us.