i guess you could rely on another established blockchain as a datasource :

we could imagine that:

1.  your wallet/client would regularly query some bitcoin nodes (randomly choosen in included hardcoded list of dozens/hundreds of existing btc nodes), waiting for a given block height to be mined by the BTC network (with X confirmations),

2. explicitely (original code from your wallet sourcecode, since it was first released) use some properties of that future btc block (let's say nNonce & nTime) as your own genesis block properties ; also explicitely for N btc confirmations for that block, to prevent accidental use of a temporary btc forked chain.

This way, everyone should use the exact same genesis data. One would have to mine it thought, this could imply some major modifications to the client, so it allows genesis block generation (meaning full startup with no block loaded, and so on)

This would require some basic extra connectivity (btc p2p) to query other btc nodes, but yeah .. that could be "fun" somehow
Again .. you'll soon have to release another version of your client, hardcoding choosen genesis block data.

This looks extremely dangerous thought .. if you really release your client long enough before genesis block gets choosen/created, malicious users would have enough time to fake the whole thing, mining their own chain at low diff, later making that blockchain the longest one .. with no hardcoded genesis block, many nasty things could be possible then

Again, this would negate your own user's confidence in your coin, right from the start i guess