Using the bitcoin-qt client on a laptop requires resyncing to the network everytime it is opened. A solution to this is to allow the client to use a remote server.
This would allow the use of the bitcoin-qt client on an irregular basis without needing to resync everytime. This would also allow the use of multiple clients, a client on the laptop and another on the phone, with them all connected to the same backend.
Did you had a look at existing RPC clients:
or check: https://en.bitcoin.it/wiki/Software#Browser_extensions
It's not BitcoinQt, but one of them might fit your needs?
Would be nice to be able to use the official bitcoin-qt client. If the GUI and bitcoind backend are appropriately separated, then it could be easy to write an RPC backend to replace bitcoind on the local machine, without changing the interface.
Perhaps have a server which acts as permanent online node and just use -connect for your client to use LAN-speed for block-sync?
As I understand it, that wouldn't do the same thing, still requiring me to resync each time. It probably wouldn't speed up syncing anyway, as it seems to be mostly CPU-bound on my laptop, I would also be running it on a remote server, not on the LAN.
I'm convinced this is a duplicate issue, but cannot find the original anywhere.
In any case, yeah this would be nice, but don't expect it to be implemented any time soon. RPC, as it is now, is not good enough for implementing Bitcoin-Qt remotely. It lacks notification support and more fine-grained queries on the transaction / address lists, and other small things. I suggest using one of the UIs that @jonasschnelli mentions.
A split at wallet - block chain daemon level is preferred to this right now (and is being worked on). It would allow more flexible setups like running multiple wallets from one block chain source.
This problem is solved by using one of the existing SPV wallet clients such as Multibit and Android Wallet. You can have them connect to a single trusted node that you run yourself, if you like.