Is it possible to add a transaction input to a transaction in the mempool? Is it possible to add an input to a transaction that is in the mempool?

For example, if someone is paying 1 BTC to their friend, I can get the raw hex data of this transaction from the mempool before it is mined.

I can then sign an unspent transaction (Let's say 3 BTC) with my own private key, list it as an input, and get the raw hex data for this. Then, I can add a 3 BTC output to someone else's address (who I want to pay), and re-broadcast this new transaction to the network.

What is stopping this from happening?
Short answer: no, this is impossible.

Longer answer: some transactions allow changing the inputs used (ANYONECANPAY inputs). It is also possible to have inputs which do not sign the outputs being created (SIGHASH_NONE). However, a transaction where all inputs are of this type, are worthless, as anyone could change the outputs to credit themselves instead.

Typically, when normal SIGHaSH_ALL signatures are used, pretty much everything of a transaction is signed (which input coins are used (and in which order), what output scripts are created, what value to assign to them, ... Everything except the signatures themselves really (which is what causes involuntary malleability).

If an input signature is marked ANYONECANPAY, it does not sign the other input coins being used. This means you can create a transaction meaning "i want x and y to be paid, but i don't care who provides the funds to do so". If all signatures in a transactioon are ANYONECANPAY, you could add extra inputs (or remove existing inputs. Just increasing the input funds would just burn them as fee, though.

On the other side, there is SIGHASH_NONE, which makes a signature not sign the transaction's outputs. This may seem to allow what you want to, but in case every signature is SIGHASH_NONE, the treansaction is completely unsafe, as anyone could replace where the outputs go.

So, even if a combination of those would in theory allow what you suggest, it is not true for typical tramsactions, amd would in fact be completely unsafe.