Disclaimer first - I am the project manager of webtrees, and was previously the project manager of PhpGedView, from which it forked at the start of 2010....
You have three real choices.
(1) a desktop application
(2) a web-based application (under your control)
(3) a web-based application (managed by someone like ancestry)
I'd be tempted to steer away from (3). Most make it very difficult to apply proper sources/citations to your research, and genealogists tend to get pretty obsessed with their sources.
I tried ancestry once, but found it very limiting. For example, it only allows you to enter "simply connected" trees, so if any of your ancestors married their cousins, you cannot link the common ancestors. It is also difficult to add sources that do not come from ancestry itself.
If you are going to publish on the web, privacy is pretty important. In some countries, privacy laws apply only to living people. In others, privacy extends for a certain number of years after death. The online services tend to operate with the privacy rules of their host country - which may be different to yours. So, check what options are available before signing up to any provider.
This leaves (1) and (2).
Whatever you do, pick an application that can read/write to the (de-facto standard) GEDCOM format. Bear in mind that many applications will either extend the specification or lose some data when saving to it, so interoperability is rarely 100%.
Web-based solutions offer the obvious advantage that the whole family can work on this together. You'll get far greater commitment from the rest of the family if they can update it diretly, rather than send updates to a central person for data entry.
Even if you use a desktop application for your main research and data entry, you'll probably still want a web-based application to publish it.
A web-based system also allows you (presumably the geek of the family) to maintain the site, perform backups, etc., while allowing your (presumably less IT literate) family members to do the fun part - researching your history.
For all its faults, ancestry.com does have a huge amount of data. So, buy your relative a subscription, and set them up an open-source, web-based system on your favourite web-hosting provider.