Trying to interface with mainstream Internet, as it exists today, on an interplanetary ship won't work. Services people make these days expect constant bidirectional communication with relatively low latency (timeouting after mere seconds). On a mission like this, my first thought would not be Netflix, but Pirate Bay.

You mean "The most resilient torrent website in the Galaxy"? The TPB crew was ahead of the times.

Easily solvable with something like proxy server on earth. Also i don't think it's smart to directly connect spaceship to the internet, so proxy should already be there in the original design.

Still, this would break parts of the internet. Some protocols between clients and servers simply depend on timing.

As a simple example, consider a websocket connection and a client that tries to reopen the connection whenever it hasn't received a ping from the server in the last X seconds.

You wouldn't speak those protocols. You'd run your whole web-browser on Earth, and then the Earth<->Mars connection would be a minimum-bandwidth delay-tolerant DOM-event synchronization protocol† to allow you to see what that web-browser is rendering, and tell it to go click things/type things. Essentially like a graphical thin-client protocol (X11 or RDP), but the primitives would be HTML/CSS elements and binary assets in their native formats, rather than native controls and assets in decompressed texture format.

† I believe the Chromecast already operates on some variant of such a protocol. It'd just need to be modified to expect events to come from both sides, rather than only from the "sender" device. (And to remove any/all timeouts and keep-alive pings, of course.)

...or, you could just use https://mosh.org to shell into any random server and run shell commands (including, say, w3m. But even things like IRC would work this way!)

They could have a proxy that would buffer requests and received data and then use the delay tolerant protocol to send that to spacecraft I suppose?