Sentimental value is worth nothing to you, if you want it, offer what you think is fair, and don't worry about insulting them. Why should something that means something to them mean anything to you? The bike is only worth what somebody is willing to pay.
Dealing with used toys, cash is king. I went with a friend who wanted a road bike, I found her a '87(?) Celeste Bianchi Limited with Columbus Steel and Shimano 600 parts (105 shifters), they asked $400 for it, and told us another buyer was interested in it sight unseen and was willing to pay $420 for it. She was only willing to pay $350, we just said we're here, and here's what we'll pay. Guess who got the bike? This bike too had a ton of sentimental value, wife spent 20 minutes saying goodbye to the bike.
Just remember, you have no attachment to their stuff and they may have a history with their stuff, but they're willing to get rid of it, so it doesn't really mean all that much to them if space is more important to them.