What kind of web apps would Martians accessing Earth internet need?

It wouldn't have regular routing with Earth. The latency would break everything.

Mars would have its own internet and the first key application would be a send and recieve queue system to connect earth.

This would basically be a REST api that lets you send files, receive notifications of files sent to you, and download them.

It would definitely be paid or metered. I'm sure the first one would be vertically integrated (app owner exclusively controls the communication link) but future versions would probably allow anyone to bring the link to earth... a 2 sided market.

Mars has garbage pings to earth, so interactive transfers are a definite no. You need to build offline first and (asynchronous) bulk transfers. TCP with really long timeouts could work, but you probably want something more appropriate because throughput would be really small for a long time until the congestion window opened; you'd almost certainly want to address packet loss handling as well.

Look into sneakernet compatible data transfers, because Mars latency is closer to that than anything else. I haven't studied it, but NNCP might work https://nncp.mirrors.quux.org/

You could 'just' run a Mars cluster and direct Martians there, but you'll need to think carefully about how changes from Earth users and Mars users are merged. The window for in-flight changes from both sides is quite large, and waiting for interactive writes to another planet is unreasonable.

This does mean hard work on conflict resolution; some may be solvable by the system, but many will require user input, and that user input is likely to be delayed, as they'll likely have moved on to something else. Lots of not simple UX there, but 'everyone' will need it, so users will be forced to adapt.

Local pornhub cache will take care of ~30% digital needs. Add piratebay, torrminatorr and couple other trackers and you are up to 50-70% of all traffic.

Go back to, oh, the early 1980s. Internet-connected computers are rare and Ma Bell charges exorbitant prices for long distance phone calls. But in the off hours you can dial up another computer and use UUCP (or something similar) to transfer mail and usenet posts. It may take a week but eventually your email might make it across the country. And I, for one, welcome our bang path overlords. Also: Never underestimate the bandwidth of a rocket full of quarter-inch tapes hurtling through space.

The theoretical subject (routing, etc.) you're looking for is Delay Tolerant Networking.

At the user layer, IPFS or a better collaborative, content-addressed filesytem could be used - request a movie, then watch it once it downloads in 20 minutes. Once it's cached locally, anyone else can access it instantly. Federated communication apps with state-resolution (like Matrix but more asynchronous) would allow instant local communications, while still seamlessly supporting interplanetary contact.

For timezones, see: https://en.wikipedia.org/wiki/Timekeeping_on_Mars

For networking, see: https://en.wikipedia.org/wiki/Interplanetary_Internet

There is no 'instant' messaging between Earth and Mars. Not possible.

Stateless communication with forward error correction. That's probably it. Everything else I would expect to be on local redundant storage and redundant devices. They should probably work on quantum entangled communication to cut the latency but that is probably a long ways out.

Optimistically we might have a quantum entanglement solution for this by the time we have martians. That could potentially allow for integration with the regular internet.

Rover?

https://youtu.be/06xS0LHIzU0?t=48

Cmdr Lewis' disco collection default installed on all music devices.

Reduced network latency when communicating with Earth.