Should I tell hiring committee about my interview opportunity at another institution? My situation might be slightly different from the question Should I tell other interviewers where else I've interviewed?. 

I have received an offer from my home institution (institution A), and the deadline for a response is fast approaching. I also received an interview request for a similar position from institution B, whose hiring decision may not be available before the deadline of the offer from A. Personally, I am more attracted to the position at institution B. Should I tell the hiring committee at my home institution about this interview at institution B to get an extended deadline for the offer from A? 

One potential complication is that one of the committee members at my home institution provided a recommendation letter for me. My inclination would be to be upfront and honest, but my family have different opinions.
Consider asking for an extension without giving a specific reason. People should expect that you are exploring options and that they have different expiration dates. Ask if it would be a problem if you can't make a decision prior to -date-. If they ask for more, say that you are exploring your options and expect to be ready by that date. But the details can remain private. Don't give the impression, however, that an offer would be unlikely to be accepted if another comes in. 

I can't predict, of course, whether you will get an extension. If you don't, then you need to deal with it as presented.

But assume that if they want to make you an offer then they actually want to hire you and not just fill a slot by some date. They have some incentive to deal with you, though they have other considerations as well.