We're not omniscient so we don't know the end results and have no right to judge a person because we don't know enough. So we pray for everyone to be saved. We have compassion on all men hoping they are elect.
God displays compassion and kindness in common grace to everyone, however common grace doesn't extend to salvation. Salvation only comes by way of God's election in eternity past, God's work of regeneration, and is applied by the gift of faith and repentance in the believer. If a man wasn't elected to salvation in eternity past he cannot be saved. Nor, will he ever want to be saved. He is an eternal enemy of God, a reprobate. God doesn't revoke his own decrees nor does he waste blood and energy on fighting his own decrees. If he has decreed reprobation for a man that man is damned from eternity past. God is always on God's side.
then what grounds do you have for supposing God would want you to pray to Him to save your entire family ? why bother?
furthermore , what grounds do you yourself have for trusting in Christ if Christ is only for the elect ? How do you get saved if you didn't know you were elect ? How did you get saved if there is no general "offer" of salvation ?
At least the hyper calvinists go all the way , they recognise the point that if God has no Gospel for everyone , if the Gospel is "only for the elect" they , being eager to remain consistant look to whom they preach , they actually look to preach only to the elect (sensible sinners) and they avoid at every cost sharing the Gospel with the none-elect because , as you have said , "there is no Gospel for them" ........
the inconsistancy arises where you have men preach the Gospel to everyone and believe God doesn't have something good for everyone , there you will find a massive discrepency , the same as when you desire a person or persons saved (Romans 9) all the time thinking God wants the opposite , you want them spared but you believe God doesn't ....... Jonah had the opposite problem!
and its really no good saying , "but we don't know who are elect , we don't know who to pray for , we don't know who to preach the Gospel to " because this endeavour will be met with inner doubt and no faith , and even great conflict , it really leaves the Christian in a hopeles situation where he is left stifled and his testimony , his witness is severely compromised .
to think , God wants men to preach salvation to all indiscriminately , to pray for all men (indiscriminately) but God really only wants to save the elect and only wishes to answer your prayers on behalf of the elect makes praying and witnessing look a mockery .... it makes it look like God is asking you to do something He is not prepared to do , in fact it makes it look like God who sent out evangelists is at war with himself.
I am glad you believe in "common grace" at least you haven't gone all the way over , usually Calvinists (if they are) who deny Free-Offer also deny Common Grace.
There is a direct link between common grace and saving Grace , one is used as leverage for the other ;
Romans 2