"Ever" here would mean "once", although "once" doesn't really fit the sentence. The speaker here is saying that the listener has not heard "our name" even once. At least, it is "unlikely".

'ever' refers to the fact ,that something has never happened before. In this sentence the negative sense of 'ever' ( something has never happened ) is negated by 'unlikely' (something not likely to happen). When you negate a negative statement ,it becomes positive. So the sentence means: It is not likely to happen,that you have never heard our name. In other words: It is likely ,that you have heard our name. So rather "you may have heard our name."

'Ever' means 'at any time'. Never means 'not at any time' or 'not ever'. 'Unlikely' means not likely or probable. So 'it is unlikely that you (have) ever heard our name' = it is not likely that you (have) heard our name at any time in the past/ in your life. You can rewrite this by using likely and never. 'It is likely that you (have) never heard our name.' = It is very probable that you have not heard our name at any time before now.

unlikely + ever = likely + never Geddit?

It means that speaker is verifying whether the person to whom it is spoken to has heard about the speaker. Here ever implies "have (heard our name) before"