Certainly a person may offer his own "freewill" prayers. According to Jewish law, however, he should not recite the full name of Hash-m in his private, personal prayer, because only the legislative body of the Sanhedrin has the prerogative to institute a prayer with the name of Hash-m.

The reason why we only find the term "freewill offering" being used with regard to the Shemoneh Esreh is because the Shemoneh Esreh was instituted to correspond to the offerings that were brought in the Beis ha'Mikdash, as the Gemara explains on Daf 26b (the morning Shemoneh Esreh corresponds to the morning Tamid offering; the afternoon Shemoneh Esreh corresponds to the afternoon Tamid offering; and the nighttime Shemoneh Esreh corresponds to the limbs and fats that were burned upon the altar during the night). Since the Shemoneh Esreh corresponds to the offerings, the Sages discussed the concept of a "freewill" Shemoneh Esreh in place of a freewill sacrificial offering.