In Shemona Esrei we Daven for Da'as - understanding, Teshuva, Geula - redemption, Refua - health, Parnassa - livelihood, etc. Yet nowhere to my knowledge do we Daven for our children. Is there perhaps somewhere we do? If not why not?
I heard from R' Kelemen that he asked Rav Shlomo Wolbe this question, and his response was that davening for the needs of one's children cannot be limited to the set times of prayer. It is necessary at every moment of child-rearing. The needs of a parent for their child change from moment to moment and are qualitatively different throughout the day.
This question comes from a faulty understanding of the purpose of shmonah esrei. The Rambam (sefer hamitzvos 6, and elsewhere) explains that tefilah (meaning specifically shmonah esrei) is an avodah. The purpose is to serve Hashem, not ourselves. We are having a convesation with Hashem about the ways in which we hope he will interact with the world. Even what may sound like personal bakashos (requests) are tied back into our desire to be able to serve our creator to the best of our abilities and so we ask for forgiveness or understanding not to fulfill a personal need, but as a way for us to be better avdei Hashem.
Therefore we don't daven for children, spouses, parents or any other relationship (why don't we daven for a good Rebbi?!) because our focus needs to be how we can serve God and how creation as a whole yearns to be connected to him.
The Tur in siman 112 brings an agada that says all the brachos in shmona esrei are paralleling something that happened in this nations past.
When Avraham was saved from the fire, the angels said magen avraham. When Yitzchok was tied to mizbeach, they said michayei meisim. When Yaakov came to the gates of rachamim and glorified the name of Hashem they said Ha'El HaKadosh. Etc.
The Levush in that siman explains the intention of this agada.
These historical events are as if they are happening every day and they include anything that happens under the sun because maaseh avos siman libanim (what happened to the fathers foreshows the children). We need to pray to Hashem that He should be with us as He was with them. That is why there was an enactment to pray of them every day.
It seems to me that we do pray for children in every shmona esrei. But it is we who are the children, hoping to be remembered in the zchus of the fathers. After considering this I don't think it is our place to be asking for our children as well.
The first idea is to know and understand that כל ישראל ערבים זה לזה (All Israel are mixed one with the other.). This means that in each of the blessings requesting assistance (healing, material sustenance, etc.) your children are included. This same principle applies to viduy (confession). One says the entire confession even if individually some detail doesn't apply.
Beyond that, there is a spot in the Amidah where one is permitted to make individual petitions. That is in the 'Shema koleinu' blessing just before saying, 'because You hear the prayers of every mouth. Etc.' Reference to this can be found in the Siddur Beit Yaacov by Rabbi Yaacov from Emden, Otzar HaSefarim edition, in the section on tefillat shacharit, pg. 71 at the top of the page.