It might require a protocol change (another soft-fork?):

1. The new "user voting transaction" could specify the transaction fee differently than it is done today. To be backwards compatible, such a transaction would have a difference of zero (0) between the values of the inputs and outputs. Therefore, old (current) miners would not very likely pick such a transaction (as they get no fee if they do so). Old nodes might not even propagate such a transaction.

2. The new "user voting transaction" could specify the miner fee as a P2SH output that has a dependency on the block it is included in, with certain conditions (e.g. SegWit ready). Only the miner who mined the block can redeem/spend that output if the mined block matches the "conditions" the user requested.
However, this requires a few additional/new script operations that allow nodes to validate the transaction against the block characteristics it is included in.

Not sure if this is possible at all, just a rough idea. But it would allow for a market/cooperation between users and miners - in contrast to today, where miners have the only voting possibility.

Not sure if this has ever been discussed, but I as a user would like to vote with my transaction fee for a soft-fork.

Example:
If my transaction is included in a SegWit-ready block, the miner get's the transaction fee. Else the fee is zero.

This would naturally result in that only miners who support SegWit would pick my transaction. Other miners would have no benefit in picking my transaction to be included in the next block. But it would give me as a user at least some voting power on the further development/progression of the Bitcoin network.