It's a feature that should be available on all websites. First, why wouldn't train operators want passengers to travel on a quieter, slightly later, connecting train where they are happy to do so? Second, why should passengers wanting a longer connection to give them more time to change trains be forced to book at a ticket office (which is necessary if wanting to purchase directly from the train operator, such as for promotional fares)? Virgin Trains West Coast has long insisted that the lengthy minimum connection times at some of its stations are essential (to the inconvenience of commuters), but they still aren't lengthy enough for some mobility impaired passengers and yet VTWC won't give them the option of specifying a longer connection time!
The only other booking engine which offers a basic means of trying to specify a longer connection time is WebTIS (as used by VTEC, as mentioned by wellhouse above), but its (awful) new design (not used by VTEC - yet) has removed this useful feature.
I'd also agree - and I think part of your post is getting at this - that there should be the option of specifying a lower minimum connection time (down to 5 minutes) for stations where the stipulated minimum connection time is longer than this (except certain London terminals and possibly Birmingham New Street, Manchester Piccadilly and/or Leeds), to cater to regular passengers who are familiar with the layout of the stations where they change trains and would benefit from a lower overall journey time by using earlier connections.